Quite a lot, I believe (although `a lot' is subjective).

A neat example of a property which can be read immediately off of a (finite) presentation $\langle X; R \rangle$ is the Deficiency of said presentation. This is defined to be $|X|-|R|$. Now, this contain some intriguing properties. For example, every group of deficiency greater than 2 is large (it has a finite-index subgroup which contains a homomorphism onto a non-abelian free group), while every group of deficiency 1 is infinite and if a deficiency 1 presentation has a relator which is a proper power then this group too is large.

Also, the word, conjugacy, etc. problems for presentations is an intriguing topic.