born and raised in St. Louis, haven't lived there in 10+ years, but parents are still there, so i visit once a year or so.
I'd say:
Not beer related:
Arch, Zoo and Art Museum are worth the price of admission (they are free - it does cost to go to the top of the arch).
Tour of AB is worth your time. Also, see if you can get tickets to Grants Farm.
It's not a trip to St. Louis if you don't stop at Ted Drewes for frozen custard.
I can't stand baseball, but a game at Busch Stadium is also a required event in St. Louis.
Walk around the Hill and you will find good to great Italian food and maybe some decent beer.
Beer related:
Perennial is great. Side Project is also.
I enjoyed Square One Brewing several years ago when in town. Not sure if they are even still there.
Urban Chestnut is solid brewery.
St Louis has lots of little 'bar districts' where you can stumble from bar to bar. The Hill, Soulard, the Landing, UCity loop, St Charles main street, Westport. All these places have 4 - 10 bars ranging from dive bar to music venue to Applebees type places. Take a cab there and a cab home and spend hours wandering around in between.