I'm a huge fan of camping on a trip, however, if you've never done it before, this probably wouldn't be a great time to start. By the time you try to buy (or even if you can rent) gear that you'd only use for a week, plus campground fees, I really doubt you'd find that you have much savings over staying in budget motels.

If you already had some gear, were going to be gone longer, or simply knew that you'd enjoy camping then it could easily be a good idea, but I think you'll find it is a case of more hassle than it is worth for this trip.