Yes, SLC is your best bet to fly in and out for visiting Moab. I'm not sure what you want to see in SLC, but in my opinion the other areas of the state are much more scenic than anything in the city. We did try to see Little Cottonwood Canyon near SLC one day, but it was snowing and the trail heads were not well marked. And we spent 1 afternoon seeing the Utah Museum of Natural Science, which was amazing, but would not be my choice with only 5 days.
I would do :
Day 1 - fly SLC, drive to Moab (approx. 3hr. 45min), on the way into Moab stop and see Dead Horse State Park, night 1 in Moab.
Day 2 - Arches NP - you can spend all day here easily. Bring in a lunch because they don't have concessions. If you didn't see Dead Horse State Park, you could go in after Arches, night 2 in Moab.
Day 3 - Maybe drive south to see Needles Overlook, Wilson Arch, and/or go east and see Fisher Towers. You can drive into Canyonlands, but you can't really get to the good hiking trails without an off-road vehicle. Alternately you could try and arrange a single day jeep trip where they will drive you into the area. Night 3 in Moab.
Day 4 - Drive from Moab to Torrey, seeing Goblin Valley State Park and Canyonlands National Park, night 4 in Torrey.
Day 5 - Drive from Torrey to SLC (3 hours) and spend the afternoon in SLC seeing what you want. Night 5 in SLC.
Day 6 - flight home.
If you can swing 1 more day, I would keep the initial Day 1-4 the same, but then:
Day 5 - Drive from Torrey to Bryce via highway 12 through Boulder, seeing Calf Creek Falls and Bryce Canyon NP. Highway 12 is incredible scenery. Night 5 in Bryce.
Day 6 - Drive from Bryce to SLC (4 hours).
Day 7 - flight home.
I agree with the advice to book early as that is a popular time of year.