The Divine Liturgy is considerably more “participatory” in nature than the N.O. Mass. As was pointed out, you’ll encounter numerous occasions where the congregation will bow and sign themselves with the Sign of the Cross (most notably at every mention of the Trinity) - just follow the others and have no fears of crossing yourself the “wrong” way. By all means I would encourage you to at least try the Byzantine right-to-left method, but remember that you are a Catholic in a Catholic Church, and the left-to-right Sign of the Cross that you may reflexively fall into out of habit is by no means unacceptable - no one will “look at you funny” if you do so. Same thing with genuflecting; although it is not a Byzantine sign of reverence, it is by no means “wrong.” I’d encourage you, however, to at least try it our way - a bow at the waist followed by a Sign of the Cross - after all, you are seeking the full “Byzantine Experience’” right? BTW - I personally continue to bow before entering my pew whenever I worship in a Latin Church (old habits, you know…) - trust me that I probably get more “looks” from the congregation than you would were you to genuflect in a Byzantine Church! As I said before, the Byzantines are far more used to cross-ritual visitors than the Latins.
You’ll experience much chanting back and forth between the priest, the deacon and the congregation (during the entire Liturgy, in fact). As Father Deacon Ed pointed out, don’t worry about trying to follow along during your first visit - the more you attend, the more the tones and chants will become second nature to you - no one will be expecting you to know them during this your first visit. You may want to try joining in the responses, however, during one of the many litanies with their repetative chanted response of “Lord, have mercy!” I guarantee you’ll pick up on this particular chanted response very quickly, and you’ll have many opportunities throughout the Liturgy to practice your newly learned skills in prostopinije (plain chant)!
You asked if there would be a collection - of course! This is, after all, a Catholic Church! There may be yet another opportunity in which to make a financial offering. Often times there will be a blessing at the conclusion of the Liturgy known as mirovanije, wherein the congregation processes up to Father to be anointed with holy oil on the forehead and to receive a piece of antidoron, the same bread used for the Eucharist but, in this case, blessed but not consecrated. Typically, there’ll be a basket next to the antidoron tray in which you may drop a small offering - I mention this merely as a “heads up,” to help you prepare for your experience and not feel so out of place.
Some other Byzantine terms that other posters mentioned that you may or may not know:
Tetrapod - a small, four-legged table (tetra = four; pod = foot) at the front of the church, directly in front of the iconastas, upon which are placed holy icons for veneration. You’ll see the Byzantine faithful, upon entering church, walk up to the tetrapod, bow before it, cross themselves and kiss the icon that’s on it before going to their pew or the area in which they wish to stand during the Liturgy. Many will also repeat this veneration as they pass the tetrapod on their way to receive the Eucharist. Please feel free to do so.
Theotokos - our name for the Blessed Virgin Mary, the Mother of God.
The Divine Liturgy should last somewhere in the neighborhood of an hour and a quarter - a bit longer if there’s mirovanije afterwards. Also, please be aware that Byzantine Catholics do not recite the filioque ("…and the Son") during the Nicene Creed when referencing the procession of the Holy Spirit. Since the Creed is typically chanted, chances are that since you’ll be unfamiliar with the chant you most likely won’t be chanting along. Just in case, however, I’d hate for you to blurt out “…and the Son” when no one else will be. I gather from your post that you’re a bit anxious about your visit as it is… no need to add the potential embarrassment of being the “lone cantor!”
Above all, my friend, don’t let fear of the unknown stand in the way of your soaking in the entire Byzantine Catholic experience. Remember that you are a Catholic worshipping with fellow Catholics who are very proud of their unique liturgical traditions, to the point that they will undoubtedly be both extremely understanding of any faux pas you may commit (trust me - they’ve seen visitors before ) as well as be delighted to help you out to ensure your liturgical experience will be meaningful!
Please let us know how it goes!