Sounds like you would want a hybrid or comfort bike. Do you know what brands the bike shop carries?
Giant along with most brands have this type of bike. Some have shocks, some don't. Shocks do add a little comfort but add weight and make you work a bit harder. You will want to ride a few and whichever one is the most comfortable is the one you want.
The cheaper ones will have gripshifters and probably a 7 speed freewheel, probably single wall rims also. The more expensive ones will have trigger shifters, 8 or 9 speed cassette and double wall rims. It probably won't matter which one you choose. If you are kind of heavy, the double wall rims will hold up better.
These type of bikes will give you an upright comfortable riding position. It's not my type of riding, but they are very nice for the type of riding you want to do.