What sort of processes would you run sound through to get this shoegaze type sound. Is it just distortion and EQ, or is it something else?

http://www.youtube.com/watch?v=-f7GuS9F0kM

Hate to tell you but most shoegaze/post-rock tones are the result of weeks spent tweaking racks and pedals.

To get close I'd suggest a distorted guitar doing fast powerchords, run into two chains:
Chain 1:
Delay>lowpass
Chain 2:
Reverb (wet signal only)>saturator (or very subtle distortion)
Then put both chains through another reverb at about 80% wet and EQ the loudest frequencies out.
Another big piece is the method behind a "wall of sound" which tends to involve multiple layers of similar instruments doing similar things, with slightly different effects.
Another big shoegaze trick is putting distortion after your reverb.
Really it's all just about experimentation. This is the music that got me into gear and production, enjoy it!

You'll need a Jazzmaster and a Twin Reverb.

Lots of reverbs.

Some sort of looper/sampler pedal is also common. Having immediate hands on control of chain switching and mix/wet/dry also helps. Volume pedals in various spots. I suppose doing it with your DAW and multiple layers of tracks is possible but it just loses that feeling of immediacy and randomness you might get from a chain of hardware pedals and amps with feedback.

There was a pretty active thread on a member here doing this for an album on his DAW some months back. He wanted to do it with all plugin effects, despite people telling him to do it analog/hardware. He reported back and said he got close with plugins but it just did not have that same vibe/sound/feel.