My wife, two sons (8 and 9) and I will be flying to Denver in June. We have one free day prior to our trip to Rocky Mountain National Park and one free day afterwards. What activities would you recommend in places like Denver, Boulder, Golden, Colorado Springs, etc.? Kid-friendly museums are okay, but we like renting bikes or exploring scenic stuff outdoors.

If baseball fans, check the Rockies schedule. A cheaper option would the Colorado Springs SkySox (minor league team).

These are the attractions in Colorado Springs that I am aware of:

Pikes Peak/Cog Railroad
Western Museum of Mining & Industry
U.S. Olympic Training Center & Hall of Fame
Garden of the Gods City Park
US Naval Academy

At it's simplest:
Colorado Springs has Pike's Peak, Garden of the Gods, Cave of the Winds, and a zoo.
Golden has the Coors brewery tour.
Boulder has a nice bike path along the creek, pedestrian mall (shops, restaurants), rock climbing/scrambling at Pioneer Park, nearby Eldorado Canyon, hiking at Chatauqua, and after touring NCAR (nice views), nice hiking behind the facility.
Denver has the river path, the U.S. Mint, Museum of Nature and Science (next to the zoo), ARt Museum (etc.), pedestrian 16th Street Mall (shops, restaurants).
Boulder's on the way to RMNP, lots of outdoor activities; I'd go for Boulder.

Oh Boy, geography lesson:
US Naval Academy is in Annapolis Maryland.
US Air Force Accademy is south of Denver in Colorado Springs.

and touring the academy is much curtailed since 9/11

Depending on where you are staying (which side of RMNP), a day at Winter Park might be fun. They have an alpine slide and other fun stuff at the base area... not sure when it opens though - should you be going in early june - may not be ready yet.

Also, for info on RMNP, drop in on rmnpforums.com. Absolutely the most thorough collection of information about RMNP on the web.
Great people, tons of info.

You will love RMNP. Lots of great hiking for sure.