How can I sample uniformly from a surface?

This paper may be of interest to you:

J. Arvo, Stratified Sampling of 2-Manifolds

It directly answers your question, though you may need to do some of the computations numerically depending on how complicated your surfaces are. Moreover, stratifying your samples will help the sampling look uniform -- sampling uniformly over the entire domain tends to look blotchy.

The problem boils down to the simulation of the point in a 2D domain with the density proportional to some easily computable function $f(x)$ (the area distortion coefficient of the mapping from the parameter domain to the surface). The simplest way is to simulate a random point $x$ in some square containing the domain, look at whether the point is in the domain, and, if it is, keep it with the probability $f(x)/M$ where $M$ is some number greater than all values of $f$. If the parameterization is reasonable enough, this should work pretty well. What examples do you have in mind?

Triangulate it and sample each triangle with a density relative to the total area.