The North Platte around Saratoga has some nice easy water and lots of good scenery. This is most likely the closest water to Cheyenne.
The Upper Colorado near Kremmling, CO especially the Ranch-del-Rio section has lots of opportunities for beginners, including some boat rental and instructions.
The A Section of the Green River, Utah near Dutch John has dam controlled flows and can be floated reliably all rear round. You can rent boats there as well.
Rancho Del Rio on the Colorado River - Services
Wet suit? Depends on what rivers, what kind of boat and what time of year you choose to float. Kayaks and duckies are 'wet rides' and you might want one. In a raft or dory you are high and dry for the most part. I do a lot of boating (raft, drift boat or 9' pontoon) in the CO, WY, UT and MT areas and have been fine with my gortex fishing waders, so I leave the wet suit at home. I do have knee high neoprene socks that get regular use.
The investment in a boat is significant. By the time I got all setup with the full rig I was in for $3000 on a used boat plus associated gear (PFD's, oars, bow rope, dry box, etc) later upgrades put me into $6k and plus a trailer.
There are many factors to consider on what boat to buy. Smaller boats like inflatable kayaks are a lot easier to transport than a 16' raft and frame, as a result they might get more use. I would suggest renting some boats and deciding what you like before dropping some big cash.
Find some like minded boating buddies to help you along. Look at the bulletin boards at the local raft shops, various on line message boards and local boating clubs. One of the sure fire ways to find new friends is to get a river permit! Most permits applications are sent in Jan-Feb with winners announced in March. It costs about $20 to enter and you an always return the permit if you can't use it or can't put a group together. While the group leader should have some river experience, the permit holder can be a newbie.