Hope I'm not too late to help, but....
As far as I know, the proof verifiers most extensively used in serious mathematics formalisation are the HOL systems (HOL4, ProofPower HOL, HOL Light), Isabelle, Coq and Mizar. Unlike the others, Isabelle can be configured with the logic it supports, but the only two logics with significant support are HOL and ZF. The HOL logic is classical, higher-order and polymorphically (but not dependently) typed. Isabelle ZF is classical, first-order and not typed. Coq's logic is intuitionistic, higher-order and polymorphically and dependently typed. Mizar's logic is classical, second-order and not typed (I think). All these systems except Mizar are implemented in the "LCF-style", which means that their logical soundness depends on only a relatively small kernel of source code.
Except for Mizar, these systems are all open source and implemented in functional programming languages (e.g. ML, Lisp, Haskell). Assuming you know a little about functional programming, the best way to get a good concrete understanding is to examine their source code, but this will take a huge amount of time because their implementations are large, complicated and poorly documented. However HOL Light (the principal system used in Hale's Flyspeck project to formalise the proof of his Kepler Conjecture proof) is much simpler than the rest, although the source code comments are rather sparse.
I released a new, open source HOL theorem prover last year, called HOL Zero. It fully implements the HOL logic, but is much simpler than the other systems (it only supports basic interaction, and is primarily aimed at checking proofs imported from other systems). I have gone to great lengths to make the source code and user interface as simple and easy to understand as possible, and the source code comments are very clear and extensive. Importantly, I have written a glossary that explains all the technical terminology used. So I recommend you take a look at HOL Zero if you want to get your hands dirty by looking at source code. If you want to actually run it, it requires OCaml (a dialect of ML) and a Unix-like operating system:
In my opinion, the main limitations of current systems are (in decreasing order of importance):
They take a long time to learn to use proficiently. It typically takes something like three to six months, and even then there is always more to learn.
Even when fairly proficient, you will inevitably occasionally (maybe once a day) come across infuriating situations where what appears to be the simplest of steps on paper takes an inordinate amount of interaction with the system (sometimes taking 10 minutes, sometimes hours).
They each have large gaps in their library of mathematical theory that has been established, so using them for serious work will inevitably involve developing a lot of supporting theory.
There is generally poor support for porting proofs between different systems.
- When displaying results, each (except HOL Zero) suffers from problems that can occasionally mislead the user as to what has actually been proved.