I can tell you about sites around SLC but will have to leave the other cities for other experts
. Do you have children with you?
The following will be listed backwards from your route (from North to South).
Starting at the northern end of Salt Lake is the Ogden/Roy area and the mountains up there go over Monte Cristo to near Bear Lake which is beautiful place and a very blue, clean lake for swimming. The town around Bear Lake is Laketown and they have a raspberry festival in the summer. Going southward, north of Salt Lake is also Farmington/Fruit Heights and near this area is the local rollercoaster type park and it is called Lagoon. Very fun amusement park for all ages. Downtown SLC includes Trolley Square - great shopping and dining - the Spaghetti Factory is not to be missed (mizithra cheese spaghetti fantastic). Also in that area is Hogle Zoo and Tracy Aviary. Smaller but still very nice. The Great Salt Lake itself is a once in a lifetime experience. Farther south on I-15 (the direction you'll be coming from) includes Thanksgiving Point which is fantastic for families and kids - really a great place to stop -- you will want to check out their website www.thanksgivingpoint.com. It would be a fantastic place to stop on a car trip! The higlights of this area for me, however, are the canyons. I highly recommend a canyon south of the SLC area - Alpine Canyon - a very scenic canyon you can wind around down to Provo (or up since you're coming the other direction!) which goes by Sundance - Robert Redford's festival / store / shop. The scenic route that direction has been compared to the Swiss Alps! It is beautiful. There is a lake up there that you can hike to called Silver Lake - it's a day hike and is worth the extra drive and hike.
Another thing South of Salt Lake that we love is Gardner Village. www.gardnervillage.com
Salt Lake canyons include Immigration Canyon and Parley's Canyon - you can go up to Park City (where they hold the Sundance Festival) and go to the ski resorts that area beautiful in the summer - Snowbird, etc. Park City has great streets to walk up and down in a quaint, shopping town with lots of character.
Those are the things we loved to do as a family when we lived there. If you have any other questions regarding Salt Lake City, don't hesitate to ask.
One more thing -- the Temple Square area in downtown SLC is gorgeous and not to be missed. Take a tour - it's not proselytizing but is a very informative about the members of the Church of Jesus Christ of Latter-day Saints that predominates the area.
If I were traveling through myself again, the ones I would NOT miss are:
Eat in SLC at Spaghetti Factory
Let me know if you have any other questions regarding Utah or the Salt Lake City area. On your way up in the South part of the state, Moab, Zion's Canyon, Lake Powell, Fishlake National Monument, Bryce Canyon are once a lifetime places and GORGEOUS and not to be missed. Bring your best camera!!!