I don't know I'd go out of my way to pick up I-10 so soon unless the intent was to hit some of the places Dave mentions. Your quickest route to Phoenix is probably I-85 to Atlanta, then take I-20 westbound, its terminus is with I-10 in western Texas. It's still fairly southern and doesn't go through much in the way of mountains, so weather shouldn't be an issue.
That said, Dave's route probably has more sight-seeing opportunities. I-20's major cities are Atlanta, Dallas, and maybe Shreveport and Birmingham depending on how major you want to think of major. It depends, I suppose, on how many days you have to to get to Phoenix and what you want to see on the way through.
I've driven both routes, though, and 700 miles a day's actually not too painful on either route: you pick up three hours from time zone changes on the way through, so there's a bit of extra daylight to work with. You won't get to see much that you can't see from the driver's seat of the car as you cruise by at 70ish miles an hour, but it's certainly do-able. And much easier than going the other direction.