Your question is somewhat misguided, because the study of logic is, _by definition_ , metalogic. That is, things like first-order theories are the _objects_ being studied.
The practice of proving things in a first-order theory rightly belong to the subjects those theories represent -- e.g. the art of proving things in a first-order theory of real analysis is something you learn from a real analysis textbook, not from a logic textbook. The _practical_ contribution that logic gives would be to teach you about things like how to set up the subject as a first-order theory or as a tool for defining kinds of objects (like abstract vector spaces).
What I _think_ you're looking for exists, but it's not as a _logic_ textbook. What you're looking for is basically something like an "introduction to higher mathematics" type of book that's geared towards preparing the reader for a more rigorous study of mathematics, including things like how to make rigorous formal arguments