Fort Collins is right at the edge of the mountains. Depending on what part of town you wind up in, it can be several miles of flatness before you get to the hills. Loveland (the city, not the pass) is another option, it pretty much connects to Fort Collins, and Longmont would be another option.

Places like Estes Park would be great, but summer is the tourist season, so finding something affordable then would be rough.

If you're not familiar with that area, June is pretty much still "winter" up in the mountains, and the good mountaineering season extends into September. Once school is in session, places like Rocky Mountain National Park get a lot less crowded.