>>What kind of bike would be good for me that would provide the smoothest ride?>>
Be it a mtn. bike, road bike or hybrid/comfort model...you might consider a bike that provides a more relaxed, upright riding position. This can ease the pressure on your wrists. You should avoid the "racing" geometry, one in which the handle bars are considerably lower than the saddle. Your riding position should prove to be the most important factor for your comfort. A front suspension can help with the bumps, along with a suspended seatpost or perhaps one of the spring-loaded Brookes saddles.
If you have more serious comfort issues, then you may want to consider a recumbent bicycle. They have very limited off-road abilities, but for body comfort...'bents can be a godsend.