First off, do NOT (repeat NOT) waste your time at the Grand Canyon Caverns. Rip-off, lame, roadside attraction. It caught my interest one time, and I took the long drive (it’s a bit of a hike off of the interstate). And boy was it ever disappointing.
Next, I would also recommend checking out the North rim. Far less touristy, great lodge, etc… Going north from Las Vegas, it also has the advantage of taking you right through Zion Canyon Natl. Park (you will need to drive through to get to the North Rim). Which, in and of itself, is a worthwhile trip.
The views from the north rim are beautiful, and different. Whether you go to the north or south rim (and given the time you have, only choose one - it is a long drive around), keep in mind it is high elevation. Early October will be an ideal time to visit, but it may get chilly at the rims - bring warm clothes.
If you do choose to go north, it is reasonable to do the “canyon loop”: Zion, Grand, and Bryce. Both Bryce and Zion are well worth seeing. And Bryce has the advantage of being smaller, and not as big a descent, so a dayhike is very reasonable.
Another option from the north is to head through Page, and go over to Monument Valley. It’s more of a drive-through place, and not really a “walk around” place. But it’s pretty cool to see in person, all these formations that are so familiar - you’ll immediately recognize it not only from all the old westerns, but just about every other SUV commercial these days as well 
However, if you do choose the south rim, to maximize your time, I’d go ahead and book a room at one of the many lodges on the south rim. Not much in Williams, and if you’re going to stay, you might as well have that incredible view in your backyard. There are a number of lodges/motels, and so they vary in price. If nothing else, at least stop into the El Tovar just to look around.
And if you are going to visit the south rim, head over to Flagstaff and check out this small little canyon called Walnut Canyon. No where near as big as even Bryce, but it is really cool in that you can take this trail (complete with many stairs) down into the canyon and walk through the indian ruins. It is also just a stunning canyon to see. I’ve recommended it to a number of friends after I checked it out, and they’ve not been disappointed.
One last note on Las Vegas, once the over-the-top-ness of all the casinos gets to you (and it will), there is a lesser known, but amazingly pretty place to see very close by. Take Charleston Ave. (st. ?) west all the way past the subburbs to a place called Red Rocks. Stunning mountains with, you guessed it, red rocks. Really beautiful, and you can take some nice hikes through there as well. From the strip, it is only about a 20 minute drive.
Hope this helps