I second the Frederica Matthews-Greene link.
Some quick tips. They are Christian. No goats on ropes to be dragged out and sacrificed, no invocation of the dark arts. The incense represents our prayers, and the candles the light of Christ.
The service is elaborate, and the sermon probably short. The Divine Liturgy is focused on Christ and communion, not the sermon.
If the service is not already English, there is likely a translation available. There are often changes to the "standard' service; a pew- or row-mate should be able to help you stay on track.
Some congregations will kneel at the consecration of the bread and wine. If that's too much for you, simply sit (or, if no pews or chairs, stand) quietly and reverently and wait for everyone to get back up.
Don't go up for communion; it's a sign of doctrinal unity -- not a wish for doctrinal unity. That part comes in the prayers. If you happen to be in a parish where they all go to communion by row, simply step out of line when it's your turn, bypassing communion, and take a piece of the bread -- it's not communion -- and eat it while you return to your place. That bread is called "andiron" -- "instead of the gifts" of communion. It might be offered instead at the end of the service; take a piece and eat it. It is wheat flour, bread, salt, yeast and water. That's all.
There could be "services in the service" - a memorial for someone who has died, or the "articles" - "breaking of the bread" that is *not communion, but a service of blessing and prayers for the well-being of a family or the parish. Both generate edible items (Kolyma - a wheat-berry preparation - or a sweet bread) that anyone can eat. In fact, if you take some but don't like/want it, it may be discarded for wildlife to eat, but not where it'll be trampled on. It's disrespectful to simply throw it in the trash.
Don't do anything you don't want to, and be respectful. If you aren't in the habit of crossing yourself, you don't have to, though many will at the mention of "Father, Son, and Holy Spirit" and at other times.. You don't have to light a candle or kiss an icon. You don't even have to smile
Some people are creep ed out by some of the prayers, thinking they are worshiping Mary. Only God is worshiped; the saints (including Mary) are *venerated and held up as examples for us, but we worship only God in Trinity. We do believe that the saints are conscious and can pray for us, still caring for our conditions, and we ask them for their prayers.
Often, a series of petitions ends with "Remembering our most blessed, most pure and glorious lady Theotokos and ever-virgin Mary, with all the saints, let us commend ourselves and our whole lives unto Christ our God." Turning it around but maintaining the sense of it, it says "Let us commit our lives to God, while remembering Mary and the saints" -- who committed their own lives to God in spectacular ways. It's good.
[Theotokos - lit. "birth-giver of God", often translated "Mother of God". Mary is the mother of our Lord Jesus, who was human and divine at his conception. She literally gave birth to God.]
Clothing should be conservative - no shorts, short skirts, bare midriffs or upper arms. A dress is better than jeans. But a nice pair of slacks will be fine in all but the most conservative of parishes.
Don't offer to correct their theology in the middle of the service... they've been doing the Liturgy of St. John Chrysostom for nearly 1500 years, and have it pretty well down.
Come with your mind open to the grace of God; it can be found there.