Barley's Brewhouse has a bunch of beers on tap. I don't care for their locations, but you'll find plenty of beers to try. Yardhouse probably has the largest selection, followed by Barley's. If you want local brew, try 75th Street Brewery or McCoy's in Westport. If you like imports, Charlie Hooper's has a ton of them available, though they don't have much on tap.
Agree on Harry's Country Club. Great place to sit outside and drink Ham's, Schlitz, etc.
While I agree Harry's is great for Shlitz, et al., I don't want you to think you can't drink better there. Here's a link to their beer and whiskey (also impressive) list.
I should clarify on Charlie Hooper's. In fairness, I really ought to give the place another shot, but the 110 bottled beers struck me as an amalgamation of all the beers that you could find if you hit up supermarkets in a 5-mile radius. You can get a pretty long list with Beck's, Beck's Dark, Beck's Light, Heineken Dark, Negro Modelo, Modelo Especial, Stella Artois, Foster's, etc., etc., Not to mention all the varieties of Bud, Miller, Michelob, and the like.
They carry full selections from some easily obtainable micros (New Belgium, Boulevard, for example).
But I have a clear recollection of perusing the entire beer list and finding exactly one beer that was interesting, surprising, and unfamiliar.
Which isn't to say you can't drink well from the available list, but it didn't strike me as a good place for the beer adventurer.
Good luck. Agree with Waldo Pizza...surprisingly nice and well-thought out selection. Charlie Hooper's is like the Missouri River, a mile wide and an inch deep. Lots of beers, but surprisingly few interesting beers. And sheesh, I'm not sure how to describe it politely, but I'll just say the Friday night crowd wasn't my scene.
Harry's Country Club in the City Market has an okay selection, certainly more interesting than most I've seen in town (only been here 8 months, though).
The Yardhouse is a chain with a location out at the Legends which has 100 or so beers on tap. I've not been, though.