The steepness of all the trails listed here (and others I've been on but which are not listed below) is the same. The steepest part of the trail is that part nearest the rim. This means that the end of a hike, when you are already tired, will be the hardest. Keep this in mind in planning day hikes (and backpacking trips, for that matter). 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. Taking a short hike or two on the first day is much better than leaping into a long hike and not being able to make it out!
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. BRING WATER!! Great views of the Canyon both east and west from Cedar Ridge. The trail is patrolled regularly by Park Rangers. Some mule trains use the trail. In particular, there will be one or two mule strings carrying people coming up in the morning from Phantom Ranch. Also, the supply mule trains for Phantom Ranch use this trail going down and coming up.
Hermit's Trail to Juniper Meadow (my name). The trail is reached by taking the West Rim road to the end. (See note under Drives about access to this road.) The trail head is about a quarter mile past the buildings at the end of the pavement. The meadow is about two miles and roughly 1200 feet down. Figure about an hour plus to get to the Meadow. Twice that coming back. BRING WATER!! The trail is marked as unmaintained -- which is true -- but the upper part is in great shape and easy to follow. Very steep, however. The trail is less traveled than those nearer the Village area. No mules -- a definite plus. This trail is only rarely (I was told every week or two) patrolled by the Park Rangers. I've not seen any Rangers there myself. The upper part of the trail is in a small canyon, so you don't get the grand vistas which are present on the other trails. You do, however, get great views of the cliff walls -- particularly the cliff forming the west (opposite) side of Hermit Canyon. Lots of fossilized lizard tracks in the Coconino sandstone alongside the trail.
Hermit's Trail to Dripping Springs. See the previous paragraph for how to reach the trailhead. This is an all-day hike, unless you're fast! Dripping Springs is about two miles beyond Juniper Meadow. There is, supposedly, water available in Dripping Springs year-round. Might be a good idea to check with the backcountry office first, if you plan on getting water at the Springs. Past Juniper Meadow, the trail narrows to a foot trail. Still easy to follow. It undulates some past the Meadow, but you will have covered most of the descent once you reach the Meadow. There are some great views down Hermit Canyon from the Meadow on. As you leave the Meadow you'll be crossing to the other (west) side of Hermit Canyon. BRING WATER!! Since this is a long hike, you'll also want food. Much of Hermit Canyon is shaded by the Canyon walls, so you may not be out in the sun quite as much as the other hikes. I should add, however, that I've only been on Hermit Trail in the spring and fall, not the summer. Don't know how much sun there is then.
Bright Angel Trail to Indian Gardens (and possibly Plateau Point). This is not the best trail, in terms of sight-seeing. However, the availability of water and frequent Ranger patrols make this an excellent choice for your first hike into the Canyon. The portion between the rim and Indian Gardens is all in the same side canyon, it is VERY well traveled, and frequented by mules. However, if you are up for an all day hike, the trip to Plateau Point is great. Indian Gardens is 4.5 miles from the rim, Plateau Point another 1.5 miles from there (though that part is a mostly flat trail). Indian Gardens is 3,000 feet below the rim -- the upper part of the trail being the steepest. On the plus side, there is water available at the 1.5 mile, and 3 mile resthouses, and at Indian Gardens. Check with the Backcountry Office or the Visitor's Center to confirm this -- the water at one or both resthouses is off during parts of the cooler months to avoid frozen pipes. There is always water at Indian Gardens. The trail is patrolled regularly by Park Rangers, and a Ranger is always on duty at Indian Gardens ranger station. Emergency telephones are also available at both rest houses. In Fall '94, the Park Service was constructing toilet facilities near the 1.5 mile rest house. There are also toilet facilities at Indian Gardens.
I've not been on all of the Horseshoe Mesa trail. It can be reached by driving east along the rim to the Grandview Point overlook. I've been about 1.5 miles down the trail on a day hike. The views are marvelous! The trail along that stretch was in fairly good shape; though, this is an unmaintained trail. It is a foot trail, not a mule trail, so don't expect a wide trail. I think the distance to the Mesa itself is 4.5 miles from the Rim, but am not sure that is right. I suspect it would be an all day affair to hike all the way to the Mesa and back. BRING WATER!! There is a spring reachable from the Mesa, but according to what I've read, it is a steep walk from the Mesa down to the spring. Would be easier to just carry enough water. The trail is not patrolled, as far as I know.
Another good one is available from the Rim Trail behind the Visitor's Center. A very short walk ( ~100 yards or so) behind the Center will bring you to the Rim Trail.
Yaki Point provides a good look over the South Kaibab Trail. It is reached by driving east along the rim about a mile past the turn off to the South Entrance, to the Yaki Point Road, then going to the end of the road.
The two less expensive places I can recommend are located at the Maswik Lodge and the Yavapai Lodge. Maswik is opposite the Bright Angel Lodge and set back from the rim. Yavapai is across from the general store and is reached via the road opposite the Visitor's Center. Maswik serves a variety of sandwiches, salads, and such. Yavapai has a cafeteria which serves a couple of entrees for each meal and various side dishes.
Both Bright Angel Lodge and the El Tovar Hotel have nice restaurants. The El Tovar has a great view out over the Canyon from the dining room. Bright Angel has a very good outdoor seating area. Prices are more expensive at these two (but the view might well be worth the extra cost!).
Snacks, sandwiches, and such are available at the Babbitt General Store and at the Bright Angel Lodge (which also has an ice cream shop).
Finally, there is one place of note in Flagstaff: Sizzler. It is located on the west side of the highway, a short ways north of where I-17 ends coming into Flagstaff. Thus, it is on your left as you arrive in Flagstaff from Phoenix, and on your right as you start to head out to Phoenix. They have a great salad bar, with lots of fresh fruit and salad fixin's -- just the right thing after a camping trip filled with dehydrated food!