You'll encounter 6 trainers who will challenge you to a fight, and promise a prize for winning. Once you get done with 6, there's a 7th... He said he wanted to fight you because you'd be tired... All he has a Porygon anyway, so he's not tough.

After beating them all, you'll get a Nugget as a prize... And guess who it's from? Cooltrainer Kevin! ^_^ He'll challenge you to a battle after he gives you the Nugget.

Misty's Pokemon can be easily taken out with a few Electric attacks. The only problem though is Quagsire who is Wooper's evolution. These strange creatures are impervious to Electric attacks since they are part Ground, so a Grass attack should do the trick.

After beating Misty she will give you her Cascade Badge and you should be on your way.

Head to Lavender Town by going East from Saffron City.