The home page of this site, left-hand column, contains lots of articles on gear selection, including gear lists.

There are two reasons for keeping your gear weight as low as you safely can. The first is that you will be walking lots of long-distance days, and doing that with a heavy pack is a good way to develop stress injuries that will take you off the trail. Nearly all the successful hikers of the long distance trails (PCT, AT, CDT) try to keep their base weight to 15 lbs. or less for this reason. You might want to look at some of the long-distance trail journals (especially the PCT and CDT, which have desert sections), many of which include gear lists, to get an idea of what many long-distance hikers carry. You can find these journals at postholer.com and trailjournals.com. Not all will have gear lists, so you'll need to look at a bunch. Try also to pick journals of those who finished or came close to finishing the trail. Actually, BrianLe's journal on postholer.com would be a good place to start--he just finished hiking the Continental Divide Trail from Canada to Mexico last Friday!

The second reason is that with desert hiking, you're going to have to carry large amounts of water. You need at least a gallon per day (8 lbs.) (please don't try to make do with less!), and if you're going where you'll have to carry several days' supply, the weight will multiply in a hurry. You therefore need to keep the rest of your gear light so you have room for all that water.

In some states (I don't know about where you'll be, but certainly up here in the Pacific NW) it is illegal to walk along an interstate highway. You will hate it anyway, with all those snorting trucks going past constantly, the trash and uneven ground off-pavement and (because many of them are fenced) being unable to get off the road except at exits. Brianle has already mentioned the problem of private property. Trespassing on private property could get you into big trouble, such as getting arrested.

There are a number of long-distance trails in the desert (Arizona Trail, Grand Enchantment Trail and others) which would be far more fun than hiking along roads! And since they are mapped and logged, you'll at least know where to find water, legal camping spots and resupply options. If you look in the Long Distance Hiking section here, there's a current thread about hiking the Arizona Trail (it's titled "1 Month on the John Muir" but the poster has decided to hike the Arizona Trail instead). The two sources I mentioned for online trail journals will have journals for these various southwestern trails.

I agree with the others here who say you should take short backpack trips in the desert to gain experience in desert conditions before you start on your long-distance trip.