I moved to the St. Louis area about 7 years ago. I'm still discovering new things, and haven't checked out all the city has to offer since I live 20 miles out in the boonies, but I'll chime in with a bunch of advice/links for you:
Here's a variation on a link posted earlier. It shows all the breweries, good beer stores, beer bars, and LHBSs in our fair city.
http://beermapping.com/maps/citymaps.php?m=stlouis
I give another nod to Schlafly - either the Tap Room or the Bottleworks.
I still haven't been to Square One. The Lafayette Square area is centered around a park. The surrounding streets contain not only the brewpub, but a damn good Mexican joint (Arcelia's - empanadas to die for), and
Bailey's Chocolate Bar which is usually a hit with SWMBOs - tons of chocolate confections, cocktails, and a good selection of Belgian brews.
The Delmar Loop is a section of Delmar Blvd in University City (aka U-City). It's a great part of town for hanging out and people watching. There are lots of bars (including Cicero's), cafe's, restaurants, shops, Vintage Vinyl records, and the Tivoli theatre (independent cinema). Cicero's (at the western end of the Loop) is great for their beer selection. Another good spot down the street is Blueberry Hill - arguably the best burgers in town. This is where Chuck Berry plays his monthly gig. On the eastern end of the Loop (just east of Skinker) is
the Pin-Up Bowl if you care to roll a few frames in a fun atmosphere. It's a cocktail lounge cum bowling alley and has some decent beer (Schlafly on tap, some New Belgiums in bottles). Good option if the weather keeps you indoors.
The Loop isn't far from Forest Park - one of the largest urban parks in the world! It was the site of the 1906 World's Fair and contains the Art Museum, several lakes, some golf courses, and miles of trails. Plenty of spots for a picnic or a nap. I work right across the street from the park and it's great to step away from the job and chill out for a while. I can't say if it's legal to drink a beer in the park, but let's just say that it has been done before.
Downtown isn't that great - sorry. There are a few spots where you can have fun and it's convenient to hang out there if you're going to a ballgame, but personally I'd steer you to the nearby neighborhoods of The Landing (immediately north of the Arch - lots of bars including
Morgan Street Brew Pub) or Soulard (south of the Arch). Soulard contains some of the oldest residences in town and is the home of the oldest
Farmer's Market west of the Mississippi. Good restaurants abound and I highly recommend
Norton's Cafe if you're in a cajun mood, or
McGurk's - one of the top 10 Irish Pubs in the country. And if you're in Soulard, you're just a short car ride from the A-B brewery...
Speaking of pubs, there are two great Welsh pubs in town: Llewellyn's (which has several locations) and Dressel's. In fact, they are just around the corner from each other in the Central West End neighborhood. It's fun to stumble from one to the other!
Dressel's is more quiet and subdued while Llewellyn's is very casual.
I'm gonna stop typing now but if I think of anything else I'll chime in again.
Oh - one last thing. This might be really odd to check out the night before a Van Halen show, but the most incredible ukulele player in the world will be playing on Friday the 25th. Jake Shimabukuro is his name. He doesn't sing. He doesn't have a band. He just conjures incredible sounds from his instrument - everything from fusion jazz to Beatles covers. A video of him playing While My Guitar Gently Weeps was an 'internet sensation' a few years back. When he isn't playing solo, he's a member of Jimmy Buffet's Coral Reefer Band. Just had to throw that out there - he's truly a master. Seriously - do a YouTube search... The show is at a sweet venue called Lucas School House in Soulard just a couple blocks from the aforementioned McGurk's.