I don't know exactly where you are in Northern Utah, what type of things you want to do with your family while camping, and how far is too far, so I'll go from my criteria. We usually like to be by water for either canoeing or swimming. We also like to have a few good hikes in the area. We're not big fishermen, but I do like to fish on occasion. For a 2 day camping trip I would go maybe 3 to 4 hour max.

RebelScum recommended a great place in Smith Morehouse. It's a nice reservoir, ok fishing, fun for canoeing and we go swimming near the boat ramp when we're there (chilly, but still fun). That's just a little over an hour from Orem, and probably a little less than that from SLC.

You also have a few campgrounds along the Mirror Lake Hiqhway. These are going to be a little crowded in the summer months, but they're nice if you can get a spot. Mirror Lake CG is the biggest, and most crowded. I like Trial Lake CG. Lots of fishing at any number of lakes that are short drives or hikes. There are several really nice hikes too--both longer and shorter ones. A nice short one is the Ruth Lake hike, which you'll need to drive a bit too, but it's a beautiful short hike.

You can also go along the south of the Uintas to various campgrounds. I've camped at Moon Lake, and it was ok, but I don't need to camp there again.

You can also head south and camp up on Boulder Mountain. There are a few smaller campgrounds there, as well as some really decent primitive areas where you could park your trailer (like along the road to Lower Bowns Reservoir. There's ATV riding around there, you can canoe or fish at the reservoir, go see Capitol Reef (kind of hot for camping there in the Summer, but really nice to drive through Cathedral Valley).

I have a few other really good suggestions that I can give you in a boardmail. They're not "secret" per se, but I'd rather not advertise them too much.