Riley,
You've been given pretty good information by the posters. The Melkite Community in Nebraska is located in Lincoln and serves the Divine Liturgy at St George Ukrainian Greek-Catholic Church there at 10am - every Sunday, I believe.
The priest serving there currently is a member of the Benedictine community at Mount Michael Benedictine School in Elkton - Father Daniel Lenz, who I think is a bi-ritual Latin priest, rather than an incardinated priest of the Eparchy of Newton of the Melkites.
As my sister, Anhelyna, has said, don't worry about crossing yourself in the Latin form - most all Eastern Catholics in the US are very attuned to Latin visitors and will pay no mind to it (yes, they'll recognize that you're a visitor, but that's mainly because our congregations tend to be small compared to a typical Latin parish, and folks generally know one another, at least by sight, if not personally). Expect to be standing a lot and sitting occasionally, but don't expect to be kneeling. Communion, as has been pointed out is by intinction - the Melkites being the sole Byzantine Rite Church that doesn't use a spoon. You'll receive standing, open your mouth, and the presbyter will place the consecrated bread in your mouth - it will be leavened, not the host to which you are accustomed.
The Liturgy will be served principally in English, with some Greek. No substantial amount of Arabic will be used unless the congregation is heavy with recent immigrants, which I'm almost certain is not the case for this particular mission. You will not hear the Filioque intoned, regardless of liturgical language.
There are two main Traditions in the Byzantine Rite - the Greek and Slav - differences are mainly notable in the style of chant. The Melkites employ a form of the Greek Tradition, as opposed to the Ukrainians, Ruthenians, Russians, and other Eastern Catholics from Slav nations (Romanians somewhat bridge the two Traditions, though they lean to the Slav).
Afterwards, the community is likely to gather in the church basement or hall for 'coffee and' - feel free to attend - it's likely someone will invite you anyway. The Lincoln Community is a rather diverse one as I recollect - you're likely to find more former Latins than cradle Melkites.
If you've any specific questions, feel free to ask. As TG noted, I've not posted here in a while, but I'll keep an eye out.
Many years,
Neil
I will have to ask my religious studies professor if her Melkite Priest friend is coming soon or not...hopefully.
