For Vegas departures, either air or bus, check here:
http://www.lasvegasgrandcanyontour.com/
To drive to the South Rim from Vegas takes 4 1/2 hours each way. About the same for the North Rim, buthere's only one accomodation within the National Park on the North Rim... the Grand Canyon Lodge. It's far less crowded on the North Rim, but keep in mind that some facilities shut down in mid-October.
The weather can be very cool at either Rim... dress appropriately.
Finally, here's a good overview of the diffrences between the three avialble destinations at the Canyon...
http://www.grandcanyon.com/which-rim.html
Welcome to the U.S. by the way!