I make the drive from the mid-Atlantic to California now and again, and there are two ways to go, really. I-80 and I-40. I’m starting to much prefer I-40 - for me the distance is about the same, scenery is slightly better (well, more varied, anyway), and I think the food is better - Tex-Mex and BBQ.
Some of the more off the beaten path sights and places -
Just west of Amarillo is the Cadillac Ranch. Cadillac Ranch - Wikipedia Not much there, and not much to do - get out, tag a car, take a picture, go about your way. It’s on the south side of the road, so you will have to loop around to get to it, so it may not be worth it. You can see it from the road, so you may choose to just wave at it as you go by.
Just across the NM line, at the first exit (after the welcome center) is Russell’s Travel Center http://www.russellsttc.com It’s a truck stop*, sure, but the big thing is the car museum. All the cars belong to the Russell family, and are both old and new. There will be about 30 or so, and they rotate them, so you never know what you will see. (After Carroll Shelby died, they brought in 5 or 6 Mustangs - thousands of All-American horsepower and tons of Detroit iron. It was a beautiful thing.) The also have other stuff on display. You will have to make 2 passes - one pass around the inner loop to look at the cars, one pass around the outer loop to look at the stuff. Look at the pictures on the web site, you’ll see what I mean. It’s free. The associated diner also makes a pretty good burger, and the other offerings are better than fair.
If you want something more exotic, there is Taste of India in San Jon, just down the road. It’s fair, and completely unexpected in the middle of Nowhere, New Mexico.
At mile 113, is a scenic overlook at Laguna Pueblo. It’s a pretty place to stop and look at the church on the hill, and stretch your legs a bit. There’s nothing there, no rest rooms, no vending, just trash cans, and, maybe, some of the locals selling trinkets beside the road. They probably not in January, but the view is pretty. If the weather is bad, just wave as you go by.
I took a picture standing on the corner, in Winslow Arizona, with a flat bed Ford Standin' on the Corner Park - Wikipedia . I grew up with that song. Bonus points if you have the song playing as you get near town. There are a few tourist stores right around that area for trinkets and souvenirs. If there is a good place to eat in Winslow, I’ve never found it, but I’m open to suggestions from the Teeming Millions.
I also disagree with friend Turble about Meteor Crater meteorcrater.com . I think it’s worth it. You can spend, say, 45 minutes and hit the highlights, or up to maybe 2 1/2 hours reading all the placards in the museum and doing the rim walk (only about 1/4 of the rim, really, and they may not be doing to that time of year). Yeah, it’s just a big hole in the ground, but what it is and how it came to be there is worth hearing about and seeing. Where else will you ever get a chance to get that close to a recognizable meteor crater on this planet?
You didn’t say how you plan to get from the GC to Henderson. If you loop to the north, you can pass thru Colorado City and Hildale, the (previous) home of Warren Jeffs. I’ve never been.
If you go back down south to I-40 west, you will pass by Seligman. It’s worth driving thru town - there are a few tourist stops. Eat at the Road Kill Cafe - food is a bit better than fair.
If you want the experience of driving on the old Route 66, stay on 66 out of Seligman to the west, to Kingman. It’s probably the longest, well preserved segment of Route 66, and it won’t add a lot of time on your run to Kingman, maybe 30 or 40 minutes, unless you stop an a roadside attraction. I’ve only ever stoped at the Grand Canyon Caverns http://gccaverns.com , and, well, I wasn’t impressed. Full disclosure - my standard is the limestone caves we have here in the east coast.
As far as where to sleep, understand the on I-40 from Amarillo to Kingman, hotels can be miles apart - 50, 60 miles or more. I stop at the Welcome Centers just inside the state lines, and get the hotel coupon books. These will let you know the rate you’ll pay, as well as an address to put into your GPS so you can see how far. There have been nights I’ve stoped only because the next hotel was just a bit farther away than I was comfortable with.
Food - other than what I’ve mentioned above, I tend to look for non-chain BBQ and Mexican on that stretch. I’ve not had a bad meal, and some that were good. Amarillo is your last best chance for Texas BBQ - I’ve not had really good brisket outside of Texas or Oklahoma.
*Truck stops - I stop at truck stops. They are easy off, easy on the interstate. They have clean restrooms. The associated convenience store has better than convenience store prices for munchies. Not supermarket prices, but better than 7-11. If you burn out a headlight, they have them, and anything else you may need for minor repairs. They often have a sit down place to eat if you feel the need (and sometimes, it’s good, like Russell’s). They have good turnover on their gas, so you won’t get a tank full of condensate. They have good turnover on the coffee, so it’s fresh. Truckers demand all those things, so they have to deliver.