Places to stay:
Downtown
Chesterfield
Central West End
Airport
Places to definitely stay away from:
Maryland Heights\
Creve Coeur
Places to eat are numerous depending on where you are staying
The Hill area for Italian food - It's the old Italian neighborhood back when there were ethnic neighborhoods.
Downtown, Charlie Giotto's is a good choice. The Horshoe casino has a Morten's steak house and also the Cinder House. Both a bit pricey but Cinder House has some good food and atmosphere. Reservations needed.
BBQ:
Pappy's
Bogarts
Sugar Fire
Adam's Smokehouse
Salt + Smoke
List of a lot of good restaurants
https://www.gayot.com/restaurants/best-stlouis-mo-top10-topratedfood_30sl.html
Things to do:
St. Louis Zoo (world class and it's free)
Art Museum (Free as well)
Science museum(also free) These three are all in Forest Park. If you come in the summer there is also the Muny Opera which is the largest outdoor theater in the US. Mostly musicals. There is also the St. Louis Wheel and the St. Louis Aquarium at Union Station downtown.
If you like to Gamble, there is Horshoe downtown and Casino Queen right across the river. Hollywood in Maryland Heights and Ameristar about 2 miles from there in St. Charles.
Missouri Boatanical Gardens (also known as Shaw's Garden)
City Museum for the kid in you
About 2 hours southwest of St. Louis are some really great float streams if you are into canoeing or tubing.
The arch and the museum at the base of the arch
Of course we have the Cardinals and the Blues
The Old and New Cathedrals
And some other ideas:
https://www.busytourist.com/things-to-do-in-st-louis-missouri/