The best advice i can give for a new visitor to the Divine Liturgy is simply to watch and observe, and to follow along as best you can.
Standing is the norm; at our church we sit during some of the litanies (Lord, have mercy, Grant it, O Lord), during the epistle, and the homily. Also after the homily, during the offering and the litany before the preparation of the gifts.
There is a LOT more incense used. Some find they are sensitive to the incense; either because it is stronger than that used in the Latin churches, or it’s just used more often. You may want to sit near the back, so as not to be up front near the censer and priest, if you are sensitive to smells or have asthma.
There are two entrances: The Little Entrance, when the priest processes with the book of the Gospel, and the Great Entrance, when he processes through the church with the bread and wine. During both of these, we stand, and when the priest passes by, either simply bow OR bow and make the sign of the cross. Depends on the custom of the church. Usually, during the Little Entrance, people bow, and during the Great Entrance, they bow and make the sign of the cross, at least at our parish.
The service is longer–can be up to 1 1/2 hours. Ours is one hour, generally, because our parish is so small.
Do not expect to hear instruments; traditionally, there are no pianos, organs or other musical instruments used. Just the human voice. At our church there is no choir, just a cantor leading the congregation in prayer. Most of the Liturgy is sung, not spoken. If they do not have musical booklets to help you follow along, just do as the Holy Spirit guides you.
Above all, liturgy is, literally, the “work of the people”. It encompasses all your senses, sight (icons), hearing (prayer), smell (incense) and taste (holy communion), and touch (standing!).
When receiving the Body and Blood of Christ, it will be both species, on a spoon. You do not cross yourself beforehand, because you don’t want to accidentally knock the priest’s hand! Simply walk up with your arms crossed across your chest (in the form of an X, or St Andrew’s Cross), and when you get up to the priest, just open your mouth as if you’re going to the dentist. Do NOT stick our your tongue. You do not need to say “Amen”. He will–if he knows what he’s doing–gently shake the particle onto your tongue, you close your mouth, and go on your way. Don’t close your mouth on the spoon.
This is, anyway, what our priest tells all visiting Latin Catholics. Enjoy the Divine Liturgy, God’s greatest gift to us.