What is the best place to start for someone in Computer Science with a graduate level background in linear algebra, probability theory, and graph theory?
I can't give you a comparison of different options but I can at least tell you that I (having that background) really liked Grimmett's book.
As an aside: would this have been better as a comment? I was debating with myself and could not decide.
I found Complexity and Criticality by Christensen and Maloney to be an excellent introduction. They devote a section to Percolation, another to the Ising model and the third to Self Organized Criticality.
They stress intuition and as such I find it to be an excellent introductory text, though if you want a more technical reference, you should probably look elsewhere.
And right on cue, these lecture notes by Garban and Steif showed up on the arxiv.
Akash mentioned and linked my lecture notes. I wanted to clarify that they are entirely based on a small subset of Grimmett's book but I made some effort to clarify with extra explanations and figures the parts of that book that were most opaque to me. In fact the whole purpose of putting those notes up was to provide an accessible introduction to Percolation. So, if anyone does plan to go through those notes, I would really appreciate feedback so that I can make them better.
I also wanted to add that Bollobas and Riordan's book is very good. It clarifies and simplifies some of the basic material covered in Grimmett's book. It also discusses concepts like 1-dependent percolation (which, I feel, can be much more useful for computer scientists than independent percolation) and random voronoi percolation, which is a fascinating topic, not least because it is a site percolation model whose critical probability in the plane is 1/2.
You could have a look here
EDIT I should have added that these are the lectures by Amitabha Bagchi at IIT Delhi
Lyons and Peres' "Probability on Trees and Networks" (available here) has a few nice chapters on percolation (and the whole book is a good recommendation for anyone interested in probability and stochastic processes on graphs).