SLC proper or one of the 'burbs?
If you are going to be near SLC proper then you might like to visit Gallensons on the corner of 200 South and 200 East near downtown. Lots of stuff.
Other then Gallensons we don't have alot of 'good' gunshops. South of SLC about 15 minutes will get you to Sportsmans Warehouse in Sandy, UT. Another 15 minutes will get you to Cabelas Lehi, UT.
Most of my nicer gun purchases have been the luck of walking into the right pawnshop at the right time.
Shooting is a bit more difficult. Some of the clays ranges have odd hours during the week and may be changing to summer hours now. None are very close to SLC proper but with a car easy enought to get to. I can't remember if any of them have rental guns.