Hey Gunner, glad to hear you are getting excited. The Fremont through Cap Reef is not a glamour run but it's worth doing if you are in the area.
You could put in at the campground, but its kinda manky and still moving quick at that point, I would maybe put in at the bridge just past the campground, near the old stone store. From there on down, with the exception of the big unnatural falls, is all class II+, some fun small rapids. Kinda manky in spots but still enough to make it fun. The big scary falls you can't really miss, they've cut off a stream meander to make room for UT24, and you'll know it pretty darn quick. The only thing to watch out for is brush and logs. I ran from just above the Hickman Bridge TH down to just above the waterfall in 2010 and it went well. I think my flow was close to 100 but not above it. Maybe 85-90? I'll go back and look at my notes on the flow. If you portage the falls you could make a nice run down to the Cathedral Valley road.
As for side hikes, Cap Reef is probably my favorite NP in Utah. There are some great trails both easy and difficult. The Cohab Canyon trail starts literally in the campground, and accesses some fun mini slot canyons, as well as provides access to the cap rock country. Grand Wash is ok, good hike but a bit long for how short the narrows are. Capitol Gorge is cool, especially if you scramble up into some of the side slots. Hickman Bridge is a cool short hike, and the hike up from the campground to look back up into the Fremont River Gorge [the class V section] is short and fun. At the end of the "Scenic Drive" is Pleasant Creek, which also has some fun hiking.
A lot of visitors to Capitol Reef never get out of the areas directly accessible from UT-24, but for those that go further you will be well rewarded. If you are coming in from the east [from Hanksville] you can take the Notom Road south before you enter the park, and it travels a long ways to the south (ultimately all the way to Bullfrog on Lake Powell). Down a ways on the front side is Cedar Mesa Campground, a tiny [7 sites I think?] primitive CG that you can't reserve. Good hiking in the area, including a nice reef exposed with lots of old fossils. Just a bit more past this is where the Burr Trail takes off to the west towards Boulder. If you go up those switchbacks, and then immediately turn right, you can get to a parking area for the "Strike Valley Overlook", and for Upper Muley Twist Canyon. The views from the overlook as incredible, and there are a bunch of arches along the road and trail up Upper Muley. Saddle Arch is a couple kms up the trail and is pretty awesome. Lots of little side canyons to explore, and eventually (a good trek) up a ways Upper Muley itself actually forms a nice slot.
Also, on the flats between Capitol Reef and Boulder Mountain is a beautiful area that includes Lower Bowns Reservoir and some meadows called Tantalus Flats. Both are quite nice, if you can get in. I've always thought trying to catch a window on Pleasant Creek to see if you could float all the way down through the Reef would be pretty awesome, but you'd need to catch it on the perfect day for that.
Sorry for the long post. I just love Cap Reef so much! As always, any further questions, don't hesitate to ask
EDIT- I will try to throw up a quick bit of beta and pics on my website so you can have something to go along with the info.