I do not know how well your knowledge of Catholicism and Orthodoxy is, but you might start with Monsignor George Appleyard’s
Light of the East: A Guide to Eastern Catholicism for Western Catholics. Monsignor Appleyard is a priest of the Ukrainian-Greek Catholic Church. A website where you can learn more about Eastern Catholic churches is that of the
Catholic Near East Welfare Association.
At the same time, I would encourage you to visit an Eastern Catholic church. No amount of head knowledge can replace a visit. You will note that in the Catholic Church, there are several churches
sui juris, each headed by their own bishop, who is in full communion with the pope of Rome (as opposed to the pope of Alexandria), currently His Holiness Pope Francis. Several of these churches
sui juris, namely the churches of the Byzantine tradition, share the same rite (well, okay, at the risk of sounding redundant, in this case the Byzantine rite). The Ukrainian-Greek Catholic Church is one of these churches
sui juris which shares the Byzantine rite and is currently headed by a major archbishop. The mission church I attend is in the Ruthenian Byzantine Catholic Church, another church
sui juris, which has metropolitan archbishops. The Ruthenians are sometimes known as the Carpatho-Rusyn Greek Catholics and have independent jurisdictions.
Each (or is it most ?) of the Byzantine Catholic churches has an Orthodox equivalent.
Not all Eastern Catholic churches
sui juris have an Orthodox or even a non-Catholic counterpart, however, and thus not all are Byzantine. The most famous of these is the Maronite Catholic Church
sui juris, headed by a patriarch (ignore the titles for now).
With this being said, on behalf of the forums posters of CAF Eastern Catholicism forum, we offer your our prayers on your spiritual journey.