The Holy Eucharist is 'the Sacrament of Sacraments' - and the Divine Liturgy is the heart of our life as Christians. Therefore, we can't attend the Divine Liturgy (or any church service for that matter) as it is some casual meeting with our friends.
First of all, we need to be dressed 'our Sunday best.' This excludes shorts, T-shirts, mini skirts - in short, anything you wouldn't wear in front of Japanese Emperor, should you have the honor of visiting him. Women and girls should have their heads covered with a veil. Make up should be reduced to minimum (preferably none), as well as jewelry - God sees only our good deeds and prayers as jewelry, whereas He sees jewelry as worthless baubles. Rule of the thumb is - hands and legs should be covered.
On Saturday (or day preceding the Divine Liturgy) we should attend the Vigil service, especially if we plan on receiving the Eucharist.
On the eve of the Liturgy (midnight), Eucharistic fast starts - that is, we refrain from any food, drink or marital relations (not to mention sin!) until morning. Eucharistic fast is a must if we plan on receiving the Eucharist - but do the fast even if you don't, as to take some of blessed bread which is given after the Liturgy (some churches under Russian influence even give out the blessed wine along with the bread). Should the Liturgy be performed at midnight, Eucharistic fast lasts 6 hours - from 6 PM.
In the morning, after all the appropriate prayers, we go to the church. Remember one important thing - try not to be late. Lateness, except for being plain rude, distracts others from prayer. If you're late, try as quietly as possible to find your place (men usually stand on the right, whereas women usually stand on the left). Regardless of what might be happening at the Liturgy, don't go around kissing icons or lighting candles (except possibly the central icon upon your arrival). Our boredom doesn't mean that Christ is any less present on the service.
Late arrival at the Liturgy is rude - but never leave the Liturgy before its end. Such was the way of Judas who left the Last Supper minutes after taking Communion. Of course, sometimes our presence is a must elsewhere - but in most cases it is not, even if it may seem so at the moment.
Once the Liturgy is over, venerate the Cross the priest is holding in his hand, take some blessed bread, venerate any icons you wish, maybe light a candle and try to avoid any conversation within the temple - such things can be postponed outside the sacred space.
However, all of these are externals. Remember what is the most important. Even if we do not wear flashy clothes, jewelry, make up, if we attend the Vigil and fast from midnight, if we're an hour early for the service and last till its end - these things will do us little good if we judge others, fight with our neighbour and indulge in other serious sins without even thinking of repentance. Bear in mind the words of our Savior: "Therefore, if you are offering your gift at the altar and there remember that your brother has something against you, leave your gift there in front of the altar. First go and be reconciled to your brother; then come and offer your gift" (Matthew 5:23-24).
Photo by: А.Айнетдинов, Москва, Россия.