I can recommend two excellent overviews of the process: a new one from GOP expert Keith Hennessey, and another from a few months back from Stan Collender. Both emphasize the extent to which there's no single drop dead moment when it comes to the debt ceiling, since Treasury can shuffle things around to avoid the worst consequences if Congress is slow to act.
I think that Matt Yglesias and Jonathan Chait are largely correct about the president's strategy here: both suggest that he should simply demand a "clean" bill (translated nicely by Paul Waldman); the logic is that neither side has leverage on this one because both sides more or less equally want the limit raised. Nevertheless, I think that the fallback position for the White House is to agree to symbolism: in particular, there's no reason that Democrats should ultimately balk at allowing a (separate) Senate vote on the silly Constitutional Balanced Budget Amendment that Republicans are backing. The truth is that for better or worse, the debt ceiling is in fact going to be a tough vote for some Republicans, and while that's mostly their own fault, it's still not out of line for Democrats to agree to taking a tough vote in return.
Also, if they want to attach language saying that budget deficits are very, very, very, very, very, very, very, very, very evil and so therefore it's imperative to cut taxes as much as possible...well, again, I think that Democrats should be willing to accept some symbolic junk.
At any rate, read Hennessey and Collender for why the debt limit is just a very different type of fight than the appropriations expiration we just went through -- or the second version of it scheduled for this fall.