What are your geographical parameters? Want to stick in the NW? Travel the country? What time of year will you be dirtbaggin? These questions will definitely affect cost. Lots of places have their little tricks for pretending to self-pay at the campground, get free showers, snag someone's leftover fries. If you ask about specific places you want to climb at, I guarantee people will give you specific beta you'll put to use when you arrive.
Do you own a car you'll travel and live out of? I've been 'dirtbaggin' since June and have kept my costs down a lot by not owning a car and traveling with other people. I've met plenty of folks living that lifestyle without a car (definitely gets you in touch with minimalism). The dirtbag migrates and if you're in the circuit you'll make friends and meet people traveling to the right spots. In terms of logistics, cost, and overall experience it will be really different if you're out there on your own or with people you set out to travel with.
I've traveled a lot in S. America and while living costs are low when you arrive, you'll inevitably spend more on the overall trip and climb less than if you stay in N. America.
If you're going to be on the road for awhile I recommend having some goals you're working towards to help keep things in perspective when you start to get burnt out. For me that has been a trip up to the Ruth this spring. That goal helped me stay focused.
You'll never look back on life and say "shit, I wish I didn't spend that year climbing in beautiful spots with all those fun people".
Enjoy it.