Welcome to America! You're planning a big trip. I can't really help you with the SF/LA portion of the trip, but I can offer some suggestions on Vegas/Grand Canyon.
Keep in mind - December is winter here! Vegas is a desert city, so day/night temps will vary greatly. It won't be really hot either. Day time will probably be in the 50's, with nighttime down in the 30's. To really see the sights, you'll want to spend 2-3 days. I prefer staying in one of the hotels on the strip, but there are tons of bargains just off Las Vegas Blvd. My personal favorite is Paris. If you go outside the city, check out the Valley of Fire or Lake Mead. If you drive down to the Grand Canyon, stop and visit the Hoover Dam. Quite impressive engineering and worth the visit.
As to the Grand Canyon, it will be winter, and you are in northern Arizona, which does get snow. Only the Southern Rim will be open, due to snowfall. Check out
National Park Service - Experience Your America, which is the website for the National Park Service. This will give you info on visiting the park, and making reservations (now) for mule trips, etc.