Thanks to RTD, it is possible (but not convenient) to get close to rocky mountain national park. One way is to take the Skyride from DIA to the Denver station, then a bus either Boulder or Longmont. From there, there are seperate runs that can take you to Lyons. Lyons is a small town down hill from Estes Park. If you can make it from Lyons to Estes (a short run that is not uncommonly hitchhiked) there is a shuttle into RMNP that drops you off a trailheads. You can then hike the trails to the end of another trailhead and then catch another shuttle to the next cluster of trails.
If you get hopelessly stuck, let me know. I'm in Longmont and my wife frequently drives to Estes Park for work. How many people are going with you?
To be absolutely honest, it is hard to get around here and experience all that we have available without a car. In Boulder, assuming you are >18 but <25, you may be able to rent those 50cc scooters at some places and take the bike lane up to RMNP (it would suck, but doable) or just hit all of the trails along the Front Range that are walking distance from the Boulder stops (actually, that's quite a few and even some summit runs, but no backpacking, just day hikes). Another option is offering someone a bit of $ to transport you around on craigslist. I love Indian Peaks Wilderness this time of year, less tourists and just as grand as RMNP, however, it would be impossible to get to without a ride or some massive preparation on biking it (major uphills).
If you will be staying in the Boulder area, there is the Boulder Hostel which is reasonably priced. If you are planning on camping at campsites, call ahead, it's tourist season.
Depending on the situation, I might be available to shuttle some folks to the hills.
Oh, I also wanted to give you a heads up. If you are from low altitude, keep in mind that everything up here is between 5K-14K. There are few things that are friendly to lowlanders. Regardless of physical fitness, altitude sickness can weaken anyone into having a very, very bad time. Unless you currently live at 5K+, I do not recommend trying to bike your way around here as a tourist and THEN have enough energy to go hiking.