Even if you don't have kids the STL zoo is consistently rated one of the best in the country, and as mentioned its free. Its in forest park near a great science center too. Forest Park also has a really great art museum.
If you do have kids the City Museum is a great place. Don't let the name fool you, its a big playground. laumeier sculpture is an interesting place and not too far out. Grant's Farm is worth considering. The Delmar loop is a fun place to walk and look around and grab a snack. You might be here just in time to grab a drink at Cicero's before they close down.
There's some great bbq here. The usual stuff we talk about ("mine's better...") does apply. Big names are Sugar Fire, Pappy's, and Bogart's. A short drive from downtown across the river in Belleville, IL is Beast BBQ and their reputation is stellar. IL also used to have one of Mike Mill's 17th Street restaurants but it is now closed.
Less traditional joints with great BBQ and a great reputation would be Salt & Smoke, Shaved Duck, and Iron Barley.
Outside of BBQ, Crown Candy Kitchen's BLT is legendary. O'Connell's burgers are a favorite. The Scottish Arms is great if you love scotch. Be sure to order some toasted ravioli somewhere and take a gooey butter cake home with you. Ted Drewes is famous for their frozen custard. If you like baseball, consider taking in a Cards game, STL loves their baseball.