I have not actually ridden the mules either, but I hike the canyon frequently, have chatted with a lot of mule riders on my trips, and know a couple of people who have done it.
Most people that I know enjoyed the experience. If you are not a hiker or not in great shape (or just don't feel like hiking), it is the easist way to get deep into the canyon. The guides that go with you are all top-notch and can help point out things in the canyon that you might miss. And of course, it is just kinda fun to be riding along like a cowboy through amazing scenery.
The drawbacks are that it can be a bit sore on your butt, you have to go with the mule train rather than explore at your own pace and it can be a little bit scary swaying back and forth in the saddle along narrow trails with the sense that your life is in the hands (hoofs?) of a mule. :)
I personally like to hike because I can stop where and when I want, take pictures, etc (plus it is good exercise). I would NOT recommend that you try to hike to the river and back in one day if this is your first time to the canyon. Certainly not during summer when temperatures in the inner gorge shoot past 110. If you do hike is summer, avoid being on the trail between 11AM to 3PM - it is just too hot to be enjoyable and you could be putting your life in danger (people trying to hike to the bottom and back in a single day during summer is the number one reason for medical evacuations and is strongly discouraged by the rangers).
Mule or not, I highly recommend that you and your family don't just stay on the rim. To really experience the Grand Canyon, you have to go into it a ways and get the full sense of being in the canyon with walls towering around you. Even if you don't hike or mule all the way to the bottom, there are many easier great day hikes that go part way in (like taking the South Kaibab Trail to Ceder Ridge or the Hermit Trail to Dripping Springs). Even here, take plenty of water and snacks (and sunscreen) and give yourself plenty of time.
And yes, the passenger rides into the canyon are all on mules, not horses. Some of the trail guides and non-passenger pack trains use horses, but all of the paying passenger rides are on mules because they are more sure footed with a big load.
See the link below for one person's experience on a mule ride.
Also, the park service sends people a video tape talking about the ride to people that have reservations. You may be able to get a copy of the tape to see what you (and your family) think.