Rim or Inner Canyon Mule Rides
These can leave from both the north and south rims. You can go down below the rim and back up the same day. They also have some that take you down to Phantom Ranch for an overnight stay. You'll head back again the next day by a different route.
These trips are booked up to a year in advance, but you'll likely have a shorter wait time if you want to book Grand Canyon mule rides for the winter months. The canyon interior is scorching hot in the summer, so a winter trip can be a great choice. It stays relatively warm down in the canyon, without the opressive summer heat. The south rim is open all year.
The tours are very safe, if you go by their record. Sure, it's scary when a 1,000' drop-off is yawning just below your right foot, but the Park Service hasn't lost a passenger in over 200 years! This is one of the best ways to see the canyon there is. If the wait time is too long for your vacation plans, all is not lost. Check out Papillon for many exciting alternatives.
For an extensive description of the south rim trip, see my Mule Ride in the Grand Canyon page. The Park Service also offers north rim rides summer only. You can book for either rim at the link below.
Have a great vacation!