I am in the process of moving to the Denver area. I am writing this both to help you out and for me as a summary of methods I have discovered to live financially responsible in Denver and still enjoy it. From my reading/asking around, there are endless things to do in the area that are free or of low cost. I have found the two expenses that really affect the cost of living are housing and transportation.
Housing: You can still find decent housing deals by actively scouring craigslist or touring neighborhoods you would like to live in and searching for for rent/sale signs. Allocate plenty of time for this process as these deals go fast and do not come available every day.
Transportation: The public transportation in the Denver area has gotten way better than it was 5 years ago (so I have heard from locals). If you need to get around, this can be a better option than trying to combat the masses of traffic in the area. If you live somewhere that will require a daily commute, it may be worth trying to add an RTD (bus and lightrail) pass to your contract. This will lower (eliminate?) the cost of traveling to and from work and gives you time to do some daily tasks (email, bills, read paper, read mmm...). If you can't work it into the contract, expect to pay $100-$200/month for open use of all the public transportation in the area. If you do decide to drive somewhere, do so in non-peak times of traffic which includes weekday work commutes, sporting events, and other major events in the area. I have found the roads to be completely open when I drove outside of those peak times.
Frugal Fun: Now that you have access to all of the public transportation, you can travel anywhere you want just by figuring out the correct buses/trains to take. If you want to get to the mountains you can ride the train to Golden and walk/bike to many open space trailheads. There are many breweries in the area (some will offer free tours/beers). The Denver area also has organized events going on all of the time, many of these are free or by donation. Take a look at the Denver events calendar and you will see there is never a shortage of things to do. Finally, go explore, there are so many parks and neighborhoods to explore in this area.
Food: I'l start by stating the obvious, do not make a habit of eating out. It can be hard to do as you adapt to city life as there are so many great options for food and many people have made eating out the norm. If you do want to go out, look for the specials/happy hours. With so many restaurants in the area, there is bound to be a good deal at one of them each day/night.
I hope this general advice is somewhat helpful and I hope you enjoy Denver if you end up there.