Welcome to RTA!
As Michael says, you don't HAVE to go the route suggested by a computer mapping program. That's just a suggested route! I'd suggest that you get a good USA map or road atlas and take a look at it. See which road hits your fancy. Yes, I-40 or I-10 would be logical if you are on the southern end of our state. But as he mentioned above, one isn't necessarily "better" than the other during the winter months (or even in the summer, for that matter).
My husband and I have done the length of I-40 from California to Tennessee over two trips. One trip (which caught I-40 in Flagstaff) is described right here. That was a summer trip. Then my husband took I-40 from Oklahoma City to Barstow, this past October. That trip is outlined here.
The Weather Channel website has an Interstate Forecast available for travelers. It's quite accurate. We traveled with a laptop and consulted that website each evening before we went to bed. You can also watch the TV channel, and local news stations as you travel, as most motels have a TV with either cable or satellite available.
As far as budget hotels are concerned: There's a motel finder here on this website. Also, you can pick up coupon booklets at Visitor Information Centers as you enter each state. These contain listings of motels who offer a discount with their coupon. Some of these coupons are also online, at roomsaver.com, but unless you print them out ahead of time or travel with a printer, it doesn't do you much good. Traveling in January (after the Winter Break dates) usually costs less for hotels than in summer, in MANY locations.
Donna