Sounds like you've already decided on doing a different hike but I'll just include the info so that you have it for another time.
This hike from Cathedral TH to Sunrise TH at Tenaya can be done just as easily from either direction but I'll describe it starting from Cathedral TH.
If your permit is for the next day you can stay at the Backpackers section of the Tuolomne Campground, $5 per person, no reservations needed or taken, have you wilderness permit ready to show the rangers when they check.
Walk, take the shuttle or drive two mile west from the campground to the Cathedral TH, limited parking along the road, a couple of bear boxes for any extra food.
First legal camping is 3.5 miles in at lower Cathedral lake. As Art pointed out the hike up until this point is unremarkable, but the destination most certainly is not. Lower Cathedral lakes is one of the gems of the park, the views of the surrounding peaks are second to none. One nice option would be to set up camp and then spend the day exploring the peaks without the pack, returning to camp for a swim, dinner and enjoy the alpenglow in the evening. Downsides to this popular destination is that you will have company at the lake, there are often bears that come to visit and permits are usually booked up well in advance.
Next day continue 5 miles towards Sunrise camp(it is sunrise not sunset) hang a right and another mile or two to Sunrise lakes. Again these are spectacular high alpine lakes. Middle Sunrise in particular is not to be missed, not visible from the trail, with several nice areas to camp, and a sweet island in the lake makes for fun swimming. You could day hike from here to several locations, among them Clouds Rest.
If you are heading back to the Sunrise TH at Tenaya Lake it is 3.5 miles from Sunrise Lakes, most of that distance down the switchbacks. As with Cathedral Lakes, Sunrise is one of the most popular destinations so you will be sharing the lake with others, bears do visit and the reservable TH quota fills up in advance.
Once back at the TH the shuttle stops here, so it is easy to get back to your car whether it is parked at Cathedral TH or Tuolomne Meadows.
Hope that is helpful, feel free to PM me if you need more Yosemite info, I visit often.
PS. Camping is legal and popular at both Lower and Upper Cathedral Lakes as well as at all three Sunrise Lakes.