I think that you complaint can be addressed searching for good textbooks with exercise (quite all), hints for solution (not all) and, for more recent one, support material available on the web.
A textbook that offer all this is George Boolos & John Burgess & Richard Jeffrey, Computability and Logic (5th ed - 2007); unfortunately the exposition is not (I think) fitted for practicing with proof systems.
My personal experience is that you can practice with different methods and then try to benefit from their interrelations.
For example, I've found my preferred proof system with tableaux method (see R.Smullyan, First-Order Logic (1969, Dover reprint) ) and then observe that the sequent rules are the tableaux rules written "upside-down". In this way, starting from the bottom [e.g. with the sequent : $\rightarrow A$] and applying "backward" tableaux rules, you will be able to construct the sequent proof.
Another approach that I've found useful is to practice with Natural Deduction, that is quite easy to learn. Then you can "apply" this ability to axiomatic (or Hilbert-style) systems, translating your Natural Deduction proofs into Hilbert system, that is axiomatic but "mimicks" ND [see also S.C.Kleene, Mathematical Logic (1967, Dover reprint)].
This proof system, unlike Mendelson's, uses "more" axioms; for example (see Kleene, page 15) conjunction ($\land$) is "managed" by :
$\vdash A \rightarrow (B \rightarrow (A \land B))$
$\vdash (A \land B) \rightarrow A$ and $\vdash (A \land B) \rightarrow B$
that are really introduction- and elimination-rules in "axiomatic" form.