How much time you choose to spend in Grand Canyon (assuming you're talking about the famous national park) really depends on your family composition and what you like to do.
If you enjoy scenic grandeur, there aren't many places that rival Grand Canyon NP. If you're
seriously experienced hikers and backpackers, Grand Canyon is a great place to spend several days or longer. If you're just doing the tourist "been there, done that" thing, one day is enough -- but don't even think about missing sunset! If you enjoy great national park experiences (ranger programs, Jr. Ranger for the kids, maybe a mule trip overnight to the bottom, etc) you could easily do a week there. It really depends on your interests and the degree of immersion you want.
For information about the park, go to
www.nps.gov/grca
National Park Service websites are slow and a little clunky and you have to drill down several levels sometimes to find the info you want (but NOT as bad as Disney sites!). There is a ton of info on GRCA's site, including information on lodging
inside the park...which is what we always do.
Also, understand that there are two main parts of GRCA. The busiest is the South Rim, but there is also a North Rim area that is beautiful and much less crowded. Both are equally inconvenient to get to from just about anywhere...GRCA is REALLY in the middle of nowhere, even for a national park.
Within reasonable driving distance of the South Rim, however, are the city of Flagstaff and the town of Sedona. Both have timeshares (especially Sedona, which is a destination unto itself) that are affiliated with RCI. However, as Dean mentioned RCI's website is down for "THE CHANGE" until Nov 15th, so you can't check anything until then. I don't know whether there are any good choices available through
DVC, but there should be.
There is so much to see in that part of the US, it's hard to tell you what to do. On one trip there, we spent a week, flying in and out of Phoenix. We drove 1,300 miles in seven days and visited Sedona, Flagstaff, the Meteor Crater, Painted Desert and Petrified Forest National Parks, Canyon de Chelley, Mesa Verde NP (CO), Monument Valley, and finally Grand Canyon. We did another trip: Vegas, Death Valley NP, Zion NP, Cedar Breaks NM, Bryce Canyon NP, North Rim of GRCA, and back to Vegas. Needless to say, we didn't spend much time in any one spot on either trip!
If I were doing your trip, I'd do an open-jaw trip, flying into Phoenix, going to GRCA or Sedona, then flying home from Vegas. You might pay a premium for dropping off the rental car in a different city (you might not, shop carefully), but the distances out west pretty much dictate that you do that sometimes.