Virtually no snow yet this year...but that could change drastically by Christmas. I live at 6000 feet and have only had 3 inches so far. Wherever you go, make sure you let someone know the trail you will be taking and when you expect to return. We do now have Phase 2 911, so we should be able to get coordinates for your location if you dial 911...but cell service is very spotty around Moab. Snow on the trails makes the trails much harder, but it makes following your tracks much easier. That time of year, you can be quite a ways out a trail, and suddenly run into a snow or ice-covered obstacle that, once taken, can block egress via the same route.
As for recommendations...shorter trails closer to town might be a good option (i.e. Hells Revenge, Fins n Things, Poison Spider, Metal Masher). Obviously, any snow/ice on the slickrock will just make it slicker rock.
As for don'ts, I would not recommend Golden Spike at that time of year...depending totally on the weather. If no snow, it would be great. But if something happens and you need help, it could take several hours to get to you. It's a long trail and daylight is short. I'd also stay away from trails near the mountains, unless you're into mudding. Depending on the weather, Porcupine Rim could be good or bad.
Do's: Take blankets, warm clothes, water, food, fire-starting stuff, flashlights. Plan to get stuck and maybe you won't.