Car camping -
East is good and mostly car-friendly. WAY less people than the Mirror Lake Highway area. Obviously the lakes with roads right up to them might be a bit busier but there are lots of areas with great solitude and scenery with water nearby. The Spirit/Hoope Lake area is good and you could tie that in with Flaming Gorge. The Blacks Fork area on the north slope is pretty nice too. Hard to pin point a spot because there really are so many options out there. Most of my car camping in the area has just been en route to backpacking so I wasn't really focusing on how nice it would be to just spend a weekend.
Backpacking -
There are bears in the Uintas but at high elevation in the backcountry, you aren't likely to have any issues. Still take precautions of course, but for most of the good backpacking in the 10k+ range, you don't really need to worry. I ran into a forest ranger once up there who told me specifically that the bears almost never make it above 9500' because of the availability of their usual food sources. I've seen bear scat and signs but only in the lower elevations, below 9k. There is never a 0% chance of bears, but if you camp up high and you don't sprinkle tuna juice and koolaid all over your tent, the chance is so incredibly small that I really wouldn't worry.
With that said, almost all of the good backpacking is going to put you up around 10-11,000 feet. There's good fishing about anywhere you would want to go so you can pretty much just pick something that looks nice and you'll be set. If you haven't done much up there, the Christmas Meadows Trailhead leads to some fine terrain although it is a bit busier than some other areas. Great meadow stream fishing and camping up in Amethyst Basin. Tenkara heaven...