The first stop in Moab should be the Visior's Center. It's a great one and they also have a good selection of off-road trail maps and books of different kinds, some with pictures and so on. The people who staff the place are more than happy to discuss the trails and offer lots of advice. So I would go there first.
Some of my favorite trails which are easy to drive and fun to do:
1) Long trail - Great scenery. Nice first trail to do. Relatively short and sweet. Some trails can be LOOOOONNG and take forever. After you've been wheeling for a few hours, your girlfriend is going to be giving you the long face. It's really worse when you're 1/2 way into a trail and you want to just get 'going' but you're 4 hrs to go ...at 15 mph! And there is so much more do to in the Moab area besides off-road trails, it can sometimes be a waste of time to wheel these trails which take a whole day. That's just my opinion from having done MANY trails in the area alone and with girlfriend/family.
2) White Rim Trail - Great sceneray. It's long, but lots to see and keep ya interested (IMO). Plus, at the end (or beginning depending on your direction), you can travers Schaeffer Canyon which is a 1200 ft series of switchbacks that take you from the top to the bottom. I've been all over and this is a MUST DO in my opinion. At the very least, from Canyonlands National Park, you can dip down Schaeffer canyon as far as you want to go and you should do it. It's easy to drive, heck, I let my wife drive that one and she's really proud that she'd done it. It makes you realize how powerful nature is when you look across and see how small full-sized trucks look which are literally hanging at the edge of a vertical drop-off to the bottom!
3) Onion Creek Trail - This is on the way into Moab from the interstate. This is quick and fun, more like a rough gravel road with several stream crossings. The scenery is great though. This is one I would have the girlfriend drive. Reason is, if you let her get into the action, she'll appreciate you doing more off-road driving (trust me on this). My wife started to dread the trails and once I put her behind the wheel, she was looking for more trails to go on. She didn't want to drive them, just explore more.
4) Eye of the Whale Arch - Ok, this is a fantastic place which very few people go to. It's one of those places you will never forget. Makes a great afternoon picnic spot too. From inside Arches Nation Park (you MUST explore and hike around Arches), there is a gravel road that takes you there. You'll see it on the visitor's map. Once there, get out and hike to the arch, but then hike through the inside of the arch to the backside where there is a nice 'bowl-shaped' area to rest and enjoy a fantastic piece of scenery that few ever get to. On a calm day it's like magic in this area and a great picnic spot. When it's windy, you might have to retreat under a ledge out of the blowing sand. Don't miss this.
5) Other trails - there are too many to mention. Don't try to do too many your first time.
Hiking trails - If you just off-road while you are in Moab, you're missing out on some of the best things the place has to offer. The visitor's center can give you a print-out of popular hiking trails in the area. For years we only off-roaded, but after getting bored of doing that, we started hiking some trails. We didn't reaslize what we were missing and I highly recommend. Corona Arch is one that is nice. Negro-Bill Trail is particularly nice ...there are a number of others. Of course night life and going through the trinket shops in Moab itself is fun too.
Mountains:
A lot of people don't realize that a little over 3hrs from Moab are some of the highest and most spectacular mountains in the USA, the San Juan Mountain area (Silverton, Ouray, Telluride). There are also many off-road trails worth exploring, known as the Alpine Loop. While Moab is dry and dessert, this area is green and mountainous ...with plenty of 14,000 ft peaks! I consider the San Juans and Moab the Jing and Jang of off-roading. You can make a triangle of a trip and do this area on your way back towards home, and I think you'll really like it. I like to hit both areas when possible. If you go in May, most of the high passes will still be snowed in, but you can go as far as you can and it's still fun and the scenery is mind blowing. Take some food and picnic along the way. Truth be told, you almost have to visit in July/August to cross the highest passes, and that's what we usually do. But who knows these days with the weather being like it is? I personally like Ouray to stay at because Silverton is awfully isolated, and Telluride seems snobby and way overpriced. The exception would be if you began in Ouray, spent the day on trails and overnight in Silverton, then head back north via the Million Dollar Highway on your way home (or hit more trails in other directions). In Ouray, you could relax in the natural hot springs, which I am happy to report aren't as overcrowed (usually) as some other hot springs often are (here's a trusty tip: go in the morning, then again at night because entrance tickets are good for the whole day!). Also in Ouray, there is the Box Canyon Falls, which is pretty neat to hike to and like right at the edge of town. Book your hotel a couple 3-4 months in advance because, no kidding, these small towns sell out in the summer. Sometimes there are last minute cancelations, but don't count on it. You can usually cancel the day or two before if need be ...if nothing else, use the 'kids puked' excuse. From Ouray, you can catch Engineer Pass trail and it's challenging enough for starters. There are a lot of great trails, and they're listed on the internet somewhere I'm sure. On the way home, there is decent white water rafting at Glenwood Springs. Not extreme, but enough to be a good thrill and not get hurt, especially in spring with the snowmelt.
Well, there you have it, some things to start with.. Good luck.
The next trip after that should be to go to the Black Hills (SD), Cross rural Montana, and go to Yellowstone region! I love to do that in my Land Rover! That's actually more fun for me. Did you know, that if you book far enough in advance (like 1/2 a year!), you can stay in Cabins right next to Old Faithful for like $80 a night?! It's worth doing and spending 3 nights there IMO. Otherwise, you end up paying double that outside the park and waiting in LOOONG lines to get in. Americans who havent' hiked in and around Yellowstone, are really missing some of the world's best stuff!! But, save that for your next trip. You're better off exploring deeper than wider (doing too much but not seeing much..).
I seem to have gotten into a cycle of doing my favorite places over and over with slightly different twists each time: Moab/Ouray area, Yellowstone/wild Wyoming area, Greek Islands/Peloponese, Rural China. Moab area and Yellowstone are real gems and they are sort of in our back yard. The trip getting there is at least half the fun!
