I spent 8 months living in Denver back in 2000 and 2001, so I am pretty familiar with it.
Denver is famously known as the “Mile High City,” because it is, more or less, 5,280 feet above sea level. The first few weeks in Denver I was affected by the altitude – I would get winded walking up a single flight of stairs and I would get tipsy after only two or three drinks. That’s good for the budget though.
You didn't mention what time of year you are going. That would help us make suggestions.
In Denver, the area of Lodo (Lower Downtown). All the old factories and warehouses have been turned into art galleries and restaurants and bars. The 16th Street Mall is a pedestrian mall with a bunch of shops and restaurants along it. Assuming they are still open, for food, Marlowe’s is good, and the Paramount Café has a nice patio. A free shuttle runs the length of the Mall.
I will second the recommendation to go up to Boulder. It's about 30 - 40 minutes from downtown Denver, and is a nice place with a pedestrian mall. Louisville, which is along the way, is also a nice little town.
Of the mountain villages, I liked Breckenridge the best. It's about an 1 1/2 or 2 hours outside of Denver. Nice town, great for both skiing or summer sports, or just wandering around and checking things out.
It is really easiest to drive, so if you can rent a car, I would do that.