You will be feeling the effects of the high altitude as you near the rim, and will have legs that are complaining about the effort required. No permit is required for day hikes. Overnight camping is prohibited without a permit. The Backcountry Office, reached via the road opposite the Visitor's Center, issues overnight permits and also has information about hiking trails. The Visitor's Center can also provide trail information.
Most trailheads have signs giving mileages, etc, to various points along the trails.
There are several hikes I can specifically recommend, and one that sounds good, but I've not tried all of it. The best, in my opinion, is down the South Kaibab Trail to Cedar Ridge . The trailhead is reached by driving up the Yaki Pt Road to the Kaibab Trail Head parking area. Cedar Ridge is about 1.5 miles down. Figure three-quarters of an hour down and twice that coming back up.