If you want a low budget compressor, I have this
one and it works well enough for the price. It fits perfect under the side of the rear seat and stores out of the way. Just make sure to plug it in and have it running before attaching to the tire or you will blow a fuse. Sure there are other more expensive options but that one will work well enough for MTN bikes and will add 10psi in about 6-7 min to a Truck tire. If/when you have money and room for an ARB or similar, go that direction. Just make sure to test it or any compressor before you go.
Something to do in Islands in the Sky/White Rim trail area which you probably won't do on the MTB's are additional trail off shoots. Both sides are going to see some repetition of traveling over the white rim trail to get there if you do them as a separate day trip so keep that in mind. Map.
On the Mineral Basin side is Taylor Canyon. It is a very isolated canyon and goes back 5 or so miles to rocks named Zeus and Moses. There is a pit toilet and a campground (flat spot) at the end. You can hike to Zeus and Moses if you like. The night view of the stars there is spectacular.
Just past Hardscrabble Hill is Fort bottom Ruin. You can hike out to the Ruin. Note it is on the same level you are hiking, not at the bottom by the cabin and river. The ruin hike out and back takes about 2 hours. More if you hike down to the cabin and river.
On the Schafer Trail side, Potash Road is pretty and has a couple of Colorado river overlooks. You might want to skip this as a day trip if you will be taking this as your exit from the WRT.
For more of a trail feel, Long Canyon is a fun little drive to get you warmed up to Moab and you can get a pic parked under the big rock. It's outside of Island in the Sky but its by Dead Horse State Park so that's easy to visit that before or after the trail depending on which way you go. You can also visit Canyon lands Visitor center in that same area. Both have great photos and overlook areas of the White Rim trail.
Lathrop Canyon is a fun little drive down to the Colorado River. At the end is a bathroom and picnic table. I don't know if people swim right there due to current. We just dipped our feet for a while. Again you will travel some of the WRT to get there. I think it's worth it.
Lots of trails to visit around Moab. I'll let others recommend those since there is so much to do.