In Arches, I went out the road that starts at Balanced Rock across Willow Flats. I was an idiot and did it in the evening and didn't come out until well after dark, almost got lost. Had two buttes I was using as a reference point but lost sight of them for about 10 minutes during which time a full moon came up. Changed the looks of the area completely. Just kept reading the tracks and finally made it out but was very nervous for a while as I had not carried enough food or water because I hadn't planned on anything like that. It was a fun road though and easy to follow (in the day time that is).
In Arches, I didn't run it, but the valley through Salt Valley is doable IF you can lower your tire air pressure and have a way to pump them back up.
The trail out to Gemini Arch, almost to the end. Though I probably could have made it to the end, I decided it was safer to walk the last 1/4 mile versus taking the chance on dropping down a 12" ledge. That was before I had any off-road experience but after wheelin a built Jeep Scrambler for years, I wouldn't hesitate today.
Then there is another trail which I can't remember except it goes up under a huge rock that fell down. Edit: This is Long Canyon Trail.
You can also do Shafer Trail (recommend down from Canyonlands) then go out on White Rim trail for a short way. If you started at the Pot Ash pits, you could do all of the White Rim trail in a long hard day.
Best advise is do as joloote said, get the book. It is what I used and will get it again when I go back out.
Make sure someone knows where you are going and when you are going to be back. Phone service is better than in 2000 when I was out there, but you could still be out of service area until you climb to the top.
Watch the weather, if it rains and the roads get to slick (they do very quick) just stop and wait. Within an hour or two after the rains stop, they will get much better very quick. Just make sure you aren't in the bottom of a ravine where the water will run.
There is a grocery store in middle of down town. It use to be the meeting place on Saturday morning for groups, some of which you could hook up with so you weren't out riding alone. Not sure if it's the same meet up point, but sure there is one some place in town.