I found this forum from an article on utah caving.
http://utahadventurer.blogspot.com/2008/08/local-experts-talk-about-favorite.html
As brandon put it there may be as much as 600 or more caves in Utah, most of which are wild (not toured). Going to a grotto meeting is a good way of getting to know some cavers and learn the ropes. In Utah County there is the timpanogas grotto, who meets every second tues of the month at Hansen's Mountaineering.
http://www.caves.org/grotto/timpgrotto/
Another resource for info on Utah caves is utahcaves.blogspot.com
Cave locations aren't given, but there are pictures, descriptions and some maps of some of the surveyed caves.
Also I post TR's and invitations to trips at
www.bogley.com fairly frequently. You can find people to take you out to some caves there.