We had a great time in the Southwest
We went to the GC last spring break, which is a wonderful time to visit. The bits of snow remaining highlight the terrain in a really neat way. We also saw Zion and Bryce canyons, the Glen Canyon Dam and a bit of Las Vegas and Flagstaff on our 9 day trip. That time of year it was even possible to get lodging in the historic lodges (Bryce Lodge doesn't open till April 1, however).
As far as hiking into the canyon, you can do that on your own. A mule trip must be planned far in advance as does the rafting. There are some raft trips that start in Page and go some distance down the Colorado, but wern't in operation as early a we were there. We used the trip advisor site to get a lot of help with our itinerary.
My kids were 10 and 12 on this trip and were quite bored with the 900 miles of driving but enjoyed the hiking. A stop in Page, AZ provided an opportunity for a boat ride on Lake Powell and a tour of Antelope Canyon which is not to be missed if you are travelling through that area.
The GCNP has shuttle buses that you must ride, but can be quite time consuming. THere are slean energy vehicles, I believe and are intended to reduce congestion on the park roads. Only people staying at the National Park properties are allowed to drive on them certain times of year.
I was quite alarmed at the lack of supervison of children near the rim. In fact, as we hiked one of the trails, a large rock falling (?) from above almost hit us. The high school students above might have been fooling around. The chaperone really looked stressed out trying to keep these kids from killing themselves (and others???).