"How long is a piece of string?" type question.
Hi Pfemm,
It's difficult to answer a "how long" question as you could drive the route and glance over places in 7 nights or so, but you could spend a Month quite easily in this area. I would consider at least 10 days before Vegas to get around in, and if you have more all well and good, you could spend 3 or 4 days in the Moab area alone and only really scratch the surface.
The West rim is a quite a long drive from the National park and some of the road to the rim is unmade and can be rough going I understand. You can get transported to the rim and the Skywalk, but it is quite expensive as it is not part of the National park system.
If you take the 261 to Muley point you will drive up the Moki dugway, a gravel road with some nice hairpins but comfortable [and level] enough in the Subaru. On the way is Goosenecks State park which is a short detour and although not much else there than the view of the river winding it's way around the rocks we liked it. Great place for a lunch break, but you will have to take your own.
As to where to stay, it'll depend how much time you can afford to this part of your trip.