Forest Park is bikeable, about six miles per loop. You can bike from the park, head east to the Arch and back and get to about 12 miles (+6 more at the park). Lindell Blvd is pretty decent to bike (e: I change this to "ok"), not a ton of traffic on weekends, but keep your head up anyway. Heavy Riff brewing is on the south side of the park and has/had good darker beers.

If you want to drive for biking, Grants Trail is a paved trail with few road crossings. A popular trail for runners, too, because it's protected, so you'll likely weave around people.

If you want to venture out a little, the Katy Trail is gravel and you'd also have few interruptions. There are cafes and at least one brewery on the St. Louis end of the trail (Good News). I think Good News is right about 10 miles out from a Katy Trail trailhead in St. Charles.