Finding a finite model
The first part of your question is answered by Trakhtenbrot's Theorem.
The second part is quite a large question indeed.  Depending on the relational structure you're working on, multiple solutions can be given.  For instance, if you're interested in formal languages, MSO over word structures corresponds to regular languages, and the matching logic (see this) corresponds to CFL, and thus have their satisfiability problem decidable.

You should have a look at Chapter 14 of Libkin, where nice segments of FO are proven to have a decidable satisfiability problem, according to the amount of quantifier alternations allowed.