Um, the main thing you can do is be reverent, and pick up a pew book/Missal.
To make the Sign of the Cross after the Byzantine fashion (any church labeled “Greek Catholic” such as the Melkites), you start by placing your thumb, index, and middle finger together. Then, you close your hand with your ring and pinky finger.
greekorthodox.org.au/general/images/Sign-of-the-Cross-final.jpg
You start on the forehead, go down to your navel, up to your RIGHT shoulder (as opposed to left), and then to your LEFT. Often this will be followed by a bow all the way to touching the ground with your right hand.
The Syriac fashion (for Chaldean, Maronite, and Syriac churches) is to do the same but going from LEFT to RIGHT.
Roman Catholics are the only ones who kneel on Sundays, especially during Easter. So don’t kneel unless everyone else is doing it. Some places do actually do kneeling even though it’s not called for. It’s usually just a sign of reverence and not a sign of irreverence like the lack of kneeling in the Roman liturgy.
Byzantines often go up and venerate the priest’s hands after Divine Liturgy. It just involves a quick peck on top of the hand whilst shaking. Make sure your lips are dry.
Which reminds me. Byzantines venerate icons. Here is the traditional way to do it, making a reverence. You approach the icon, and make the Sign of the Cross, bow touching the ground with your right hand, make the Sign of the Cross again, bow, and on the way up, you kiss the icon - generally you aim for the foot of the particular person in the image, and then come up and make the Sign of the Cross again and bow, touching the ground, and you’re doing. This is called a “reverence”.
Don’t wear lipstick.
You can’t receive communion, I’m guessing, but most churches don’t mind if you go up for a blessing from the priest. Just tell him you can’t receive - or stand back and cross your arms over your chest - don’t lean forward.
Byzantines and Maronites receive only on the mouth by intinction. Byzantines are served communion on a golden spoon - the bread, which is leavened, is placed into the chalice. Also, the priest or the deacon, whoever is distributing, should be told your name because they call you by name in the serving. Don’t say Amen. Lean forward, tilt your head back, and open wide.
Maronites use hosts but the priest dips them in the chalice so you have to receive on the tongue, basically Roman style.
No need to genuflect in these churches! Crossing yourself and a profound bow are sufficient signs of adoration for Christ in the Eucharist.
Can’t help with the Copts. But if you go to a Melkite Church or a Maronite Church or a Coptic Church you probably should not talk about how awesome Islam is and how it’s a religion of peace.
