As a recent transplant to Colorado, I'll second what Lsupimp said. Even after being at elevation for 6 months or so, I still get winded on some moderate-rated trails. "Moderate" is a relative term- some of the hikes around here are very strenuous and unless you're in excellent shape, you may struggle. Take plenty of water, you'll get thirsty.

I've yet to make it to Breck so I can't offer help there, but some of my favorite hikes in the Boulder area have been:

NCAR trails (especially the hike to Mallory cave) which are all relatively easy. Lots of connecting trails as well, so potential for longer hikes.

Miller rock via Ceran St. Vrain trailhead. The Ceran St. Vrain trail follows a meandering trout stream, which is a very easy hike, and then you have a steep climb up a jeep trail to Miller Rock, which has unimpeded views of the eastern plains and the Continental divide. The view from the top of the rock alone is worth the hike.

Hiking to Eldorado Falls. Accessed from the Ethel Harrold trailhead, it's only about a 500' elevation drop with a new trail that's very easy to walk along. There is also a decent loop and a few other trails that connect here so you can easily turn this into a longer hike. The drive up here from Baseline/Flagstaff road is excellent, and there are lots of trailheads along the way.

Chautauqua Park. As pimp said, there are some really good trails here but it tends to get very busy. The path to Royal Arch is awesome but may be too strenuous. If you're looking for solitude, this is definitely not the place. I usually avoid the trails here, driving a few hundred yards further to gregory canyon or heading to NCAR for my quick hikes.

Rattlesnake Gulch trail at Eldorado Canyon State Park. Eldorado Canyon has stunning views, a great stream running through it, and world-class rock climbing. The trail is an out-and-back, but has about 1200' elevation gain if you take the small loop up to the railroad tracks.
Along the way, you pass by the ruins of an old hotel, and can keep going to a scenic overlook that has great views of the divide and the plains. The trail itself isn't a spectacular hike, but the views are what makes this one.
Most of the trail has views of the climbing walls, so it's pretty neat to watch and listen to the calls of the climbers belaying up and down. Added bonus- stay till dusk and watch the climbers coming down the mountain by flashlight as you enjoy a cold local beer by the stream.