I think the Arch trams and museum MIGHT be closed due to renovations. Please check out before planning your trip.
No need to go to the Hill for great Italian. Caleco's is probably right across the street from your hotel -- Broadway and Chestnut. Or it could be 4th and Chestnut. Not sure when 4th turns into Broadway but Caleco's is great and a lot of Rams' players -- and often the other team's players -- will be there and fairly accessible to fans. The food is fantastic -- and so are the drinks.
Carriage rides will probably be ongoing in the evenings.
Mike Shannon's, at 7th and Market (downtown), and Ballpark Village -- near Busch Stadium -- are both must-see's. The PBR bar in Ballpark Village has a mechanical bull that is fun. Shannon's is another watering hold for local athletes after sporting events. Steaks are superb!!
City Museum is a hoot. Simply a lot of fun. Not far from the Dome.
Wonderful Art Museum and Zoo in Forest Park. The Jewel Box will have its annual holiday flower display available for a small fee. Missouri Botanical Garden, which is close to the aforementioned Hill, is always decorated to the nines for Christmas. Simply beautiful.
Steinberg Rink in Forest Park for ice-skating. The Fox Theater is offering "Wicked,' but I fear it may be sold out. St. Louis Symphony, though, won't be sold out, I'm thinking, if that's your thing.
Union Station is beautiful but has never lived up to expectations. Not far from the Dome. Travel west on Market Street about 8 blocks and when you see the clocktower on the left, that is Union Station. Beautiful plaza -- Aloe Plaza -- on right with spectacular fountain and a really bizarre Serra sculpture.
Speaking of sculptures: Visit Citygarden, 801 Market St., downtown, simply the neatest melding of art and gardens in the Midwest and very accessible to everyone. Really well-thought out and beautiful. Encompasses a wide variety of artwork -- something for everyone.
The Cahokia Mounds Park is fantastic. There is a great diorama and movie explaining what is known about the mound builders and you will have a chance to climb Monk's Mound, the largest remaining mound, I believe, anywhere. Lots of artifacts to examine, too.
Also: Charlie Gitto's, I believe, is another great downtown St. Louis pasta house located at 207 North Sixth St. It is THE favorite of Tony LaRussa, he of the Baseball Hall of Fame and ex-Cardinals' manager.
If you have the money, go to Tony's which I think is now located 410 Market Street, also in downtown. Not sure if still a Michelin 3-star restaurant but it held that distinction for many, many years.