The particle-in-a-box is pretty much the simplest case of the Schroedinger equation, and pretty much the place to start with for any actual equation solving. Unfortunately, I can't point to any specific basic book, my library kind of has this stuff spread out amongst many different books, and none of them are good introductory texts. It is a partial differential equation, although when you consider one dimensional particle-in-box cases, that is basically an ordinary differential equation. A lot of the work is in getting the boundary conditions right.
Once you start looking at realistic systems (e.g. the hydrogen atom), it gets a lot more complicated pretty quick.
As far as the Bohr model goes, you should know that Bohr cheated. His model is more or less bogus. It DID give the right energy levels for hydrogen, and as one of the first serious quantum mechanical models, it has considerable historical importance. (People had to treat the crazy quantum ideas seriously when somebody calculated the hydrogen energy levels.) It is also mathematically easy. You hear about it a lot for those reasons, but don't be fooled into believing it too much.