For a peasant family, the primary source is usually the parish register. For a Cossack family, things are a bit more complicated and interesting at the same time. Cossacks of the Hetmanate were recorded separately from the pospolite (peasants)—there were special lists, computs, and registers of hundreds and regiments. After the annexation of Left-Bank Ukraine to the all-Russian system of government in the late 18th century, many of these documents were compiled into the so-called Rumyantsev inventory of Little Russia from 1765–1769—a unique source unavailable in most other regions and often a key point for reconstructing family trees beyond the 18th century.
Another important detail: some Cossack elders were included in the Russian nobility in the late 18th and early 19th centuries, and confirmation of this lineage was achieved through noble parliamentary assemblies. Accordingly, family information can be stored in genealogical books and nobility records, not just in church or Cossack collections.
If we're talking about a peasant or bourgeois family, the search is more traditional:- parish registers and confessional records;
- census records of the relevant period;
- census and family records from the second half of the 19th century.
It's worth remembering that class boundaries in Little Russia were not as rigid as in the Great Russian provinces, and a family could change class over time.
The administrative history of the region also influences which collection to search for documents.