I'm not familiar with that Hilton, but it's probably a fine hotel. No idea about the restaurant. You are very convenient for walking to Laclede's Landing, which has a few bars that cater mostly to younger crowds. There is also one good restaurant - the Drunken Fish, along with a few okay places. Lumiere Place is also on the Landing and all of their restaurants have received good reviews, although they are pricey.
As you probably know, you're also an easy walk from the Arch and the Old Courthouse. Both are worth visiting.
Good downtown restaurants within easy walking distance include Charlie Gitto's and Mike Shannon's. A good place for breakfast is Rooster. A short cab ride can take you to the Broadway Oyster Bar (cajun - live music - a lowdown fun place). You could also cab to Mosaic (maybe 12 blocks from the hotel, so I guess you could also walk). It's close to the City Museum, if you're going there. I also like to recommend Lafayette Square - Sqwires, Square One, 1111 Mississippi, Ricardo's and Vin de Set are all good restaurants there, Bailey's Chocolate Bar is a fun bar.
Not really much shopping around downtown. What did you have in mind? If you like craft/antique shops, make your way out to Main Street St. Charles. You might also enjoy exploring downtown Kirkwood. Cherokee Street has several antique stores and South Grand / Morganford also has some fun shops and restaurants.
Lots more - anything specific you have in mind? Any questions about attractions?