This page is to analyse the relationships between fathers and children in Dragon Ball Z/GT (although I'm focusing more on DBZ): why do they act this or that way with their kid, how does the paternal behavior affect the child and so on.
Goku&Gohan -
Goku&Goten -
Gohan&Pan
Vegeta&Mirai Trunks -
Vegeta&Trunks -
Vegeta&Bra
Krilin&Marron -
Satan&Videl
Goku is the first character of Dragon Ball to become a father. Being the loving guy we know and probably influenced by Chichi's very strong maternal instincts, he is very close of his first son, Gohan, even naming him after his Grandpa (in the anime they say the baby reacted at the mention of rice but... Manga, people. Manga.)Gohan, as the four year old nice little mama-boy we first meet, is full of love and admiration for his father, proving it enough during the battle against Radditz. Already, a strong bond units the two characters, but it still is the 'usual' one between father and son. As the story goes on, they develop a warrior kind of bond, especially during the Cell saga where Goku litterally hands the destiny of Earth to his son. Later on, when Gohan is a teenager, we see how he misses his father and how happy he is to see him again. As his father, Gohan always keeps something childish in his character and this side of him is more obvious in such occasion, when he seems to be a little boy devoted to his beloved dad again. I think that the bond between them was so strong that he felt as if a part of him was missing when Goku was gone. It partly explains why, as a child, he began to like Piccolo so quickly. With his father dead, the kid needed a father figure and since the Namek showed a few signs of care, Gohan immediately adopted him.
Those two are probably closer than any other father and child, for they fought side by side many times and have similar minds; happy going, confident, hopeful and goofy (although Gohan is much more less than Goku). Goku often proved he was a caring and loving father depiste his warrior duties, such as promising to go fishing after a battle, but as time passed he begun to consider Gohan more as a fighter than as a child, which was pointed out by Piccolo during the fight against Cell. Goku wanted the monster to pull Gohan's potential out by beating him out, but forgot Gohan didn't want to fight and counted on him to help him. As Piccolo said "He's not like you, he doesn't enjoy figthing! He's just a little boy wondering why the father he trusts so much isn't coming to rescue him!" (approximate translation from French manga #34). I think this marked a point where Goku stopped putting too much on his son's shoulder and kept wanting to protect him despite him growing up. He was aware more of his paternal responsabilities than at the time of the Namek Saga, for example, where he knew Gohan could take care of himself and that Piccolo and Krilin were there to help him.
What is mostly known about Goku and Goten is that Goku was absent during Goten's prime childhood. Instead, Gohan was a sort of father figure for his brother and Goku seemed to be completely absent from his second son's life. The latter barely reacted when he was announced he would meet his father and instead of being excited and happy, he even hid from him! During the boo Saga, although the circumstances aren't for a nice family reunion, Goten remains distant toward his father and Goku accepts it, knowing it is no time to discuss such things. However, when it is time for him to return to the After World, we see that after all, Goten does care about his father and doesn't dare to ask a hug from him. Goku takes him in his arms and offers us one of the cutest scenes in Dragon Ball Z. After all those events though, we find the two after a little spar session, Goten grown lazy and fights-despising. Although we can see Goku likes him and wants to spend time with him, it is obvious they are not as close as Gohan and Goku are.An obvious reason for this is that, Goku being absent for a major part of his son's life, they could never develop the same bond than he did with his first kid. They met when Goten was old enough to decide by himself what was good or bad and in his mind, Gohan was the model to admire and try to imitate, not some strange, goofy, unknown man.
However, they managed to not become ennemies, for the simple reason that they are both too good in nature to hate a member of their family, especilly if he happens to be quite the same as you are. And it it obvious they are quite alike, not only of face but also of mind. Although Goten isn't as wild and fight-found as his dad was, we do recognise Goku in him.
The very first time we see Pan, she's four year old and already able to fly alone around the world at full speed. Full of life and energy, she's quite the opposite of the shy and peaceful child Gohan used to be. She's more like her grandfather Goku, since she loves fighting and seems like the type of child to get in troubles easily yet not worry too much about it. In Dragon Ball GT, she follows Trunks and Goku in space in the sake of adventure and is proven to be a typical tomboy teenager.Gohan loves his daughter and learnt enough from himself to let her participate in a world martial art tournament at the age of four. Proud of this freedom, Pan isn't used to be told what to do, such as staying on Earth when a spatial Dragon Balls quest is possible. She has grown in a world of peace and isn't as aware of danger as her father was at the same age. She's not half as mature as he was, nor as found of school as he was. Pan is the true descendant of Saiyans, seeming to live for fighting. However, she inherited of the love of life and happy-going personnality of her father and grandfather. She also possesses the tricky mind of her mother, managing to get what she wants in whatever way is needed.
Despite his more concern and responsable mind, Gohan raises his daughter quite the same way Goku raised him: with liberty and trust. Far from giving a strict direction in life or telling her what to do all the time, he allows her to experience life by herself as he was allowed (or obliged?) to do.
Mirai Trunks has never known his father for he died in the hands of the androids before his son was old enough to remember him. He has heard of him from his mother, who brushed up a not so nice portrait of him. However, Mirai Trunks was still eager to meet his father and felt naive love for him, for the only reason he was his son. The desillusion is cruel and followed by a struggling of feelings for the time traveller. He hates Vegeta for being heartless and self-centered, yet loves him and wants him to be proud of him.Vegeta doesn't like this new Super Saiyan at all, especially that he doesn't look a bit like a Saiyan. He slowly changes his mind about him, especially after deducing Mirai Trunks is really his son. Although his saiyan pride forbids him to show any affection, his thoughts prove to us he's beginning to like the young man. However, the best proof of his love for his undirect son is his reaction to Cell's murder. Then he reveals the hidden feelings and the pain caused by the loss of a loved one, which is revealed to be, as he claimed all his life long, to be a mistake. But the proof is there: Vegeta couldn't handle the death of Mirai Trunks. When he hears that, the latter is rather pleased and glad. Finally, when he has to return to his timeline, the two Saiyans trade a smile and a move of fingers.
Mirai Trunks leaves with the confidence his father had good in him. I think Vegeta learnt more from the boy, since he couldn't deny the existence of love within him anymore.
Strong of his experience with Mirai Trunks, knowing his son can become a strong saiyan warrior, Vegeta accepts more easily the existence of little Trunks. I don't believe he ever acts as a loving caring friendly father with him, but he does love him. When Trunks is eight, he begins training him (it surprises me he didn't begin before... Bulma?) and is surprised to see the kid can already become Super Saiyan. He promises to take him to the amusement park; is beaming with pride when Trunks defeats Goten at the tournament; knocks him down to save his life (against Boo)... Vegeta loves Trunks, in his own way. They share their first hug when Trunks is eight, which highly surprises the child, unused to such familiarity from his father. I believe that after that adventure, Vegeta showed more affection, or less rudeness, to his son. Ten years later, we discover how he's naturally disappointed by Trunks' lack of interest in fighting.Trunks grew up in an easy going world, in the richest family on Earth, with Bulma as a mother. He had everything he wanted, whenever he wanted it. He's a spoiled brat and thus does not seek for Vegeta's love as Mirai Trunks did. However, being congratulated by his father is important to him, as well as showing he's the best in everything (Saiyan heritage; little Trunks, having grown up with his father, is more aware of it). Therefor he accepts and is proud to be trained, then to defeat his best friend when he wouldn't have forgiven to be beaten. Trunks is a purple haired/playful version of Vegeta: he's rude to those who annoy him, has pride enough to fill Capsule Corporation and hates to admit he's wrong. However, he's but a child and naive to many things, such as the true nature of his cold, unaccessible father, which explains his surprise when he's hugged for the very first time. He doesn't understand why his father is acting so weird suddenly. Later on he swears to avenge him and shows more similarities with the Saiyan Prince, losing himself in fights and unable to give up. After that fight though, the world becomes peaceful again and his interests change. Vegeta is back to the stone warrior he's known; they don't get any closer of each other. However, Trunks grew up during this fight and knows better than thinking his father is heartless. I don't think he ever thought so, but now he's certain it is not the case.
Bra is the typical daddy's girl. No matter how Vegeta acts cold and angry, she will get what she wants from him. She manages to take him shopping with her and to shave his mustach. What I believe to be the sources of this apparent submission is the high ressemblance between Bulma and Bra: Vegeta recognizes his wife in his daughter and has greater tolerance for both. Also, Bra is his princess, the opposite of what he is. He is violence, rudeness, loneliness (up to some point), grief and sorrow. She is light, joy, laughter, friendly and carefree. He wants to protect her 'purety', thus is an over-protecting father.She grew up in a world empty of ennemies such as Freezer or Boo. She never learns fighting and it is told she knows flying... But obviously she's neglecting her saiyan heritage. How can Vegeta accepts it, if it was such a crime for Trunks to neglect fighting? Well Bra is so alike Bulma and Vegeta wouldn't imagine his wife on a battle field. She's never learnt the necessity of martial arts and never really gets to use it. She lives among a group of undefeatable warriors in which Vegeta is the second in strength and probably experience as well. Finally, it seems her potential, although probably high, never gave any sign of existence on which Vegeta could play to get his daughter to train. And Bra, being her mother's daughter, has NO interest into getting hurt, dirty and sweaty. Hard work is just not her thing.
On the other side, her dislike for Vegeta's reason to live doesn't stop them from being pretty close. Bra naturally loves her father for the simple reason she loves everyone, especially those who give her lots of attention. And Vegeta sure gives her plenty.
Krilin is a simple, caring man; he naturally becomes a good loving father. Seeing how Marron is a perfect match of him and C-18 probably solidified an affection that existed from the announcement of C-18's pregnancy.Krilin has always been an insecure character and often in the series did we hear him complaining about dying single, wanting to seduce women or such things. Having a wife and a child is probably more significant to him than to any other character. Basically, he realised his lifetime dream. Therefor he's inclined to want to protect his family even more than Goku or Vegeta would. Marron is his ultimate success: him, the noseless ex-monk midget considered as weak by the aliens, has one of the most lovely daughters ever.
Marron has the sweet nature of her father and doesn't ask a lot to life. She's not into fighting despite the powers of her parents and is happy by staying on a tiny island with them and Roshi. Spending so much time with so little people helped to develop strong bonds between them, especially Krilin, as the openly father he is (opposed to C-18 who tends to cool down her feelings), and Marron. Since ever Krilin has been taking good care of her, playing with her and protecting her more than her mother probably did. She trusts him, loves him, admires him with the calm certitude of those who lived with the obvious since ever: Krilin will always be there for her. As growing up, she learns more about the realities of life, but Krilin has done all that's possible for her and she's still tenderly loving him. They have the easiest realationship in the whole series: everything is simple and natural with them, yet strong and undeniable.
Those are probably the less close father&child of the series. Satan is far too much into himself to care about anyone else and Videl feels superior to even her father: they are distant and seem to have little in common. But that's looking to the surface.Videl's mother died a long time ago. Combining this to his nature selfish nature, you may understand why Satan isn't letting himself getting involved into a true relationship with someone who probably reminds him a lot of his dead wife. Although the circumstances of the death are unknown, it might explain why Satan is so protective of his daughter, as in not allowing her to date 'weak' guys. He knows she's strong and feels a great pride about it, but in the same time he refuses to admit it (his reaction at the news of her beating up during the Tournament proves it). She's his little daughter and he's the world's savior: he has to protect her and she cannot handle life on her own. His love for her is proven when he learns about her death: tears and threats aren't spared from him.
Videl, since her father didn't really spend quality time with her, learnt to defend herself. From her emotionnal solitude mixed with the invasion of medias into her life, she built up a shell around her feelings, a shell made of scorn for medias and thus for her father who's so into that game of fame. She fights and trains to prove him she can live on her own and doesn't need his 'protection'. She doubts about him saving the world and mocks him. But inside, she does love him. She has simply known to hide her feelings all her life long and isn't really given a chance to express them.