there are a lot of combinations you could drive, but esentially you are looking at splitting at birmingham and heading up to memphis and getting on I-40 or dropping town to I-10 and hitting new orleans. I-10 will get you new orleans and then west thru houston and then san antonio which has the alamo and the river walk and tower of the americas, etc. then its a long desolate drive thru west texas where you can angle up north and shoot into new mexico around hobbs and that will put you in range of carsbad caverns and roswell and then go up to albuquerqe and either go up a bit more and see santa fe or go west thru arizona where you will go thru the painted desert and drive by meteor crater and then to flagstaff and up from there to the south rim.
I-40 will take you thru memphis which has some cool stuff to see and then thru little rock and over to ft. smith and then on to oklahoma city, amarillo, then at albuquerque and youre heading west thru arizona.
I-10 doesnt have camping that i know of, we always stop at rest areas. there is a nice big welcome center/rest area an hour before you get to new orleans and other nice big one as soon as you get into texas...after that, there are a decent number of just small rest areas along the way, most dont have rest rooms, just parking.
I-40 will have some camp grounds. we were going west from little rock and there were several state camp grounds along the way and we stayed at one that was right on a lake. after that, a few camp grounds and then once past oklahoma city going thru amarillo, its back to rest areas.
not sure how old your son is, but you are saying a 'baby', so you might not be wanting to rough it like you would if you didnt have an infant...not sure if rest areas would be OK with you having a baby. might just have to spring for a room a few nights until you get on location. arkansas is about the only state you'll go thru with nice state camp grounds.
if youve never been to new orleans or san antonio, it might be worth it to take 10 and check them out and to break up the trip some if you like. 40 doesnt have much other than memphis and rest of that route is scenery, not a lot of getting out and doing much. either route will take you the painted desert and that is a national park that you drive thru and is really cool. i'd definitely suggest doing that.