Definitely head south. You are not far from Sundance and it is a BEAUTIFUL drive there. Plus, it is a real "touristy" place....very fun!!
There are some great malls downtown.
If you are in a "learning" mode, check out Temple square. Sometimes the Tabernacle choir sings, and I think you can go in, but you can hear them as you walk around the square. There are also statues that tell the history of the Mormon church...I am not a believer, but it is very interesting.
Also, I think their big summer "thing" if "founder days" (I think that is what it is called), so just be aware if you are going during that time..BIG Mormon party!! My best friend is a Salt Lake native. If you have any specific SLC questions, pm me and I will call her and get her answers!! If I remember correctly, Founder's day is near the end of June.
Weather is great in June....it will be hot during the day (high 80's to low 90's), but cool off into the 60-70's at night. Obviously the earlier part of June is cooler, and those temps are for the city itself. When you get out into the wilderness areas it is much cooler.
Have you ever been there before? Landing in Salt Lake is such a trip....the mountains are RIGHT there!!! ....and, their HUGE!!
I love that city!!!
Beca