Great things to do in the winter:

Museum of Nature & Science -- see the museum, IMAX, or both
16th Street Mall -- lots of shopping
Art Museum
Ocean Journey -- an aquarium
Not sure how nice the Botanic Gardens are in winter, although they always have some great indoor exhibits.

Great places to eat:

Vesta Dipping Grill or Sambuca -- upscale places in the LoDo neighborhood
Denver Chophouse -- low key, also in LoDo
240 Union -- upscale place in Lakewood
Watercourse -- great vegetarian near Capitol Hill
There are some great ethnic places around town, too.

For proximity to all this, seems to me the best place to stay would be downtown. Hope you have a great trip!