Such a twist - I moved to Madison last year from Columbia. If you like Madison, you'll find it easy to fit in in Columbia. You'll also find that the winter doesn't last six months! I can deal with the cold but the crappy weather seems to drag on and on here...
I prefer the near west side of Madison and I prefer the south side of Columbia, if that helps you get an idea of the vibe. Grad students are more dispersed in Columbia - but I'd say you should only live super close to campus if you don't have a car. I take the bus and bike in Madison but I drove everywhere in Columbia. As a grad student, you can easily get a parking spot in one of the garages on campus.
There is no real Madison "near east" equivalence in Columbia. There's East Campus - but that's undergrad territory. Undergrads also love all the Brookside buildings - and other apartment complexes that offer shuttles to downtown and campus.
I rented at Katy Place, Walnut Brook and DBC rentals while I was in Columbia. I liked them all. No matter what you choose, you'll get a lot of bang for your buck in CoMo.
And while the microbrews aren't quite up to par with what Wisconsin has to offer, do try the beers at Flat Branch.