Adeodatus: Welcome! I am also a Traditional Catholic, and what I found was that the Byzantine Rite was even more traditional, even more beautiful, and breath-taking, and heavenly than what I had seen in the Tridentine Mass. So I stayed here. There are struggles with liturgical abuses in the East as well, but except when they are said by biritual Roman-rite priests (which is often a disaster) the abuses are on a totally different order than they are on the West.
That being said, the Byzantine Rite IS different than the Roman Rite, and a lot of the hostility coming from Byzantine Catholics towards the “Latin exiles” that come to our Liturgy is because people from the Roman Rite often do not take the time to research our customs and traditions in order to see how it is supposed to be done - instead they take with them a lot of Roman customs which don’t belong in the East, and which they often insist on retaining.
For example, standing and kneeling have different significances in the East and the West. Your kneeling is our standing; we stand not out of lack of reverence for Our Lord (as one perceives in the Novus Ordo - I will kneel for Communion whenever I have to go to a Novus Ordo) but because of respect for Him. Kneeling has a strictly penitential meaning to it and is forbidden by our canons (those of the Quinisext Council, I believe - its canons regarding faith and praxis are authoritative in the East) on Sundays. So if you go to Divine Liturgy, please do not kneel. It IS appropriate to make a minor prostration (bowing from your waist and touching the floor before making the Sign of the Cross) at the epiklesis (when the priest says “Changing them by His Holy Spirit. Amen, Amen, Amen” - note that this prayer which completes the act of “transubstantiation” occurs AFTER the Words of Institution, unlike in the Roman Rite) if you do not have pews in your way.
Secondly, when you enter a church, what we before taking our place in the pews (if your church has pews - traditionally, the church wouldn’t) is not the Eucharist which is behind the iconostas but the icons. So, instead of making a genuflection, it is appropriate to make a deep bow from your waist. Old Believers will make a minor or even a great prostration when entering a church, and then another one as Liturgy starts. I make a single minor prostration before kissing the icon on the tetrapod (the four-legged table in front of the iconostas) - you should always venerate that icon before going to your pew.
Finally, just be respectful of that fact that a lot of traditions you may be used to - statues, kneeling, the Rosary, the Stations of the Cross, Latin, Low Masses, confessionals, churches stripped of their iconostases - do not belong in the East, and a lot of resentment towards Tridentine Catholics comes from the fact that before Vatican II a lot of these Latinizations were imposed on the Byzantine Rite. Vatican II was as good for us as it was disastrous for you (and actually, the process started with Pius XII before Vatican II). The generally liberal climate of Vatican II helped us gain acceptance, ironically enough; it is unfortunate that most Tridentine Catholics still regard it is a deviation tainted with “the errors of the Greeks” simply because it is non-Roman and non-Tridentine. This isn’t because it’s liberal; aside from a couple of translation issues in the Revised Divine Liturgy, it’s as Traditional and “conservative” as you could possibly want. Heck, we haven’t even introduced such liberal innovations as the ORGAN yet!
Brother David: Old Church Slavonic is our sacred language. It isn’t “traditional” for us to have Liturgy in the vernacular - I’ll believe that when you convince the Old Believers.

The fact that Catholics have clung to Slavonic long after most Orthodox gave it up is because we are closer to the Old Believers than the Russian Orthodox are.