Ukrainian genealogical research lacks the huge resources that compliment using the vital records of many other places, especially New England or Québec. For those areas there are published indexes, compilations and abstracts, and many published family records to help the researcher. Alas, not so with the Ukraine! Bear in mind that Ukrainian genealogy didn’t even exist 30 years ago.
For anyone with Ukrainian ancestors there are some tools that you should become familiar. Above all, there’s one indispensable guide, John D. Pihach’s Ukrainian Genealogy: A Beginner’s Guide (Edmonton: Canadian Institute of Ukrainian Studies, 2007). (Despite its subtitle, I doubt that you’ll outgrow it no matter what your eventual level of expertise!). Pihach takes you clearly through the topic step by step, starting with records in North America (with a focus on Canada), then through vital and land records and where you can find them. Since the book was published, many more records, now located in both the Ukraine and in Poland, have been scanned and can be searched on line.
I was fortunate in that all of my Ukrainian roots run through a single municipality and I’ve known the name since I was a boy. Even so, I found a gazetteer helpful in identifying places. Unfortunately, Brian J. Lenius’s Genealogical Gazetteer of Galicia (Anola, Manitoba: The Author, 1999). no longer seems to be for sale. If you can locate a copy, it will help you identify all Roman Catholic and Greek Catholic parishes in Galicia (the Western Ukraine) and alternative names.
Another starting point is this convenient guide to Ukrainian genealogical sources.