Almost nothing can reliably be said about a group just from a presentation in finite time. (In fact, the abelianisation is just about the only thing one can reliably compute.) Most strikingly, there is no algorithm to recognise whether a given presentation represents the trivial group. More generally, one cannot in general solve 'the word problem' - ie, there is no algorithm to determine whether a given element is non-trivial. See Chuck Miller's survey article for details.
(Update. I inserted the word 'reliably' above in deference to Joel David Hamkins' fair comment. (Update 2. I then inserted the phrase 'in finite time' to be strictly correct, in an effort to head off further argument.) It is true that, in many special cases, there is information that can be read off from a specific presentation. This is more or less the topic of combinatorial group theory! But I want to emphasise that you can do nothing with an arbitrary presentation.)
On the other hand, there is a growing realisation that, surprisingly, if one is given a solution to the word problem (by an oracle, say) then one can compute quite a lot of information. Daniel Groves and I proved that, in these circumstances, one can determine whether the group in question is free. Nicholas Touikan generalised this to show that one cam compute the Grushko decomposition.