Bear Mountain is a nice hike. It is fairly strenuous. Still, you can always take breaks and take you time. The climb up through Fern Canyon is a staircase but beautiful. jpb is right to climb up Fern Canyon to the peak then walk around the backside back down through Bear Canyon.
An easier hike in the Flatirons is to go to the Royal Arch. You would start from Chattaqua park which is just off of Baseline road against the foothills. Not as much climbing, though there is still plenty. Still it's a more gentle hike.
Overall though I'd advise to go higher. The Flatirons are in the Foothills and it's still pretty darn warm around here. The temps are more pleasant in Rocky Mountain National Park. The trails off of the Bear Lake trail head in the park will take you to gorgeous mountain lakes and easy to hike to peaks like Flattop mountain.
Another choice is the Indian Peaks. Nice hikes start from the Hessie and 4th of July trailheads. Plus you don't have to pay a fee as you do with the park to get in.
For either start fairly early. These are popular spots. Parking gets to be an issue fairly quickly on a weekend. I'd say try to be there by 8am. Make sure you bring food, water, and some extra clothes. Something for rain and to keep you warm. Another reason for the earlier start is to beat out thunderstorms. They do happen on a regular basis in the rockies, generally around 1pm is when they'll start. Try to be off any peaks or high ridges by then.