Here in Colorado, there are peaks called "fourteeners." These are peaks higher than 14k ft. Some are pretty simple hikes. Some are scary. Some are on public land. Some are on private land (so you'll get in trouble if you don't get permission).
http://en.wikipedia.org/wiki/Fourteener
http://www.14ers.com/ (disclaimer: a previous co-worker helped/helps run that site).
One piece of advice is to get a hunting or fishing license (if you're hiking in Colorado). Part of the fee is for "mountain rescue." If you never need it, then that 25 cents of the "habitat stamp" goes to waste. If you decide to be cheap, then you get billed actual costs (think price of a car), if you need to be rescued.
One of the problems that may (or may not) bother you is the altitude. My boss was fine at sea level - able to do decent times in triathlon events. But it took him a year to get accustomed to Denver, and he was huffing and puffing like a life-long cigarette smoker. The first month he was here, he'd get out of breath going up a flight of stairs.
Pick something easy for your first. Don't decide you want a technical climb (where you need gear). You might find that you get altitude sickness and can't go above 9-10,000 feet. Many folks who drive up Pikes Peak find that they need someone to drive their car back down, because they're so incapacitated that they can't even drive.
