Your group is the fundamental group of a 2-dimensional orbifold with underlying surface the 2-sphere and 3 cone points of order p.  It follows that it acts properly discontinuously and cocompactly by isometries on the 2-sphere when p=2, the Euclidean plane when p=3, and the hyperbolic plane when p>3.  Hence it's infinite when p>2.

UPDATE:

Pete Clark's answer explains the details that I outlined very eloquently.  I'd just like to add a couple of further remarks.

 - To determine which sort of geometry (spherical, Euclidean or hyperbolic) an orbifold $O$ admits (ie upon which space your group acts as a discrete group if isometries) you just need to look at the (orbifold) Euler characteristic, defined to be

$\chi(O)=\chi(|O|)+\sum_i (1-1/p_i)$

where $|O|$ is the underlying surface and $p_i$ are the orders of your cone points.  So we see that this is positive when $p=2$, zero when $p=3$ and negative when $p<3$.

 - For a nice introduction to 2-dimensional orbifolds, I recommend Peter Scott's article [The geometries of 3-manifolds][1].


  [1]: http://www.math.lsa.umich.edu/~pscott/8geoms.pdf