Peel back Peale's summit
It is remarkable Mount Peale is as easy to summit as it is, because it is an impressive 12,721' high. It rises over 8,000' above the surrounding deserts in Moab, but you drive so high up that the climb is not that hard at all. This is sort of off trail, but sort of on trail. Mostly, it's an unmaintained social trail, but getting to that point may require some off trail travel. Due to the isolation of the peak, I expect you will have the summit to yourself.
If you've come to hike Mount Peale, you will probably drive to La Sal Pass. You can either leave from the pass, or hike back about 1/2 mile on the road to a two track road numbered "4725". If you choose to leave from the pass, hike up towards Peale and a large prominent gully to the west of the main body of the peak. If you choose to hike down the road, do so and then take FR 4725 past the cabin and continue up hill to the two track's rough end, and look for the cairns and social trail which drop out of the forest below the large gully. Either way, you're looking to get below the gully, as the talus covered slopes are not fun and you want to get up on the ridge in a safe manner. The gully offers the best and most worn option.
Once in the gully, you can see cairns and a fairly good social trail which you will hike up. While not totally obvious, this trail takes you all the way to the ridge line. About 2/3 of the way up, the trail reaches the alpine vegetation and leaves the gully, heading up the slope northeast towards the ridge. You can take this trail, as while steep, it does get you to where you want to be, or make your own way up the slope to the ridge.
Once on the ridge, there is a noticeable trail which continues across to the summit. It passes just below the false summit and more or less goes all the way to the wind shelter on the true summit at 12,721'. Views are quite good, but western ones are blocked by the mountain to the west, Mount Tukuhnikivatz. You can either descend the way you came, or make the class 3 traverse to the Mount Tukuhnikivatz. That mountain's east approach face is far less traveled and the talus harder to get by, but worth the effort.
WARNING! Hiking and outdoor related sports can be dangerous. Be responsible and prepare for the trip. Study the area you are entering and plan accordingly. Dress for the current and unexpected weather changes. Take plenty of water. Never go alone. Make an itinerary with your plan(s), route(s), destination(s) and expected return time. Give your itinerary to trusted family and/or friends.