In reply to blue_sundown:
The easiest and best way to do it on the cheap is in a van / car you can sleep in, as you can stay in many of the lift carparks and lay-bys. Camping isn't officially allowed in many areas, though if you're discrete, get the tent away early and leave no trace no one really minds.
Corvara, Arabba, Selva, San Cassiano, Cortina, Badia, La Villa are all ideal places.
Mike that's on the forums frequently has some apartments available near the Marmolada:
http://www.casa-alfredino.co.uk
With regards to routes there are so many to choose from, I'd suggest working out which bit you want to visit first.
For a Dolomites overview see:
http://www.ukclimbing.com/articles/page.php?id=2708
I hope that helps!