I'm not really a pool player, but I like to hang out at Billiards when my friends are playing. Not a lot of dining area since it is primarily a pool hall, but they do have the 2nd best burger in Springfield.
It's called the Hustler Club (as in pool hustler). Eat one and wash it down with a cold beer.
The blue room is a small meeting space that is available for rentals. Scotty will rent it out to just about anyone (as long as you don't scare the people there to play pool).
Billiards is a pretty standard bar/pool hall with the exception being its surprisingly good food. The burgers and fried mushrooms are a personal favorite.
Another nice thing: the jukebox has a wide selection, so you'll inevitably see country folk being forced to listen to gangster rap or thug nation types listening to Johnny Cash.