Algebraic stacks from scratch

Another good place to look are the notes of Master's course on stacks by Betrand Toen. I think they pretty much do exactly what you are looking for.

Here's the quick summary: You will want to read section 1 of Cours 2 where the term geometric context is defined. It's basically a category with a Grothendieck topology with a fixed class of morphisms that you call geometric. The main example are commutative rings with the etale topology or the smooth topology. This induces coverings in the presheaf category in the standard way.

Then skip straight to Cours 5. Although you said that you are comfortable with descent this section is definitely worth a close look. It introduces a homotopy theory on the category of groupoids and shows that there always is a weakly equivalent groupoid such that your functor becomes strict. It then reformulates descent via homotopy limits. The upshot is a nice category of stacks, Definition 4.4.

Then jump straight to Cours 8, Definition 1.4. and you've got algebraic stacks. The only point where you will need schemes or algebraic spaces is for representable morphisms, but judging from the remark after the definition you can get around that as well.

You should read the following post: http://math.columbia.edu/~dejong/wordpress/?p=8

It partially explains why this approach was not taken in the stacks project, and probably isn't generally taken elsewhere.

Now I'll just quote "... any full discussion of the theory of algebraic stacks is going to mention affine schemes, schemes, and algebraic spaces. It will still be the case that the most interesting objects of study are algebraic varieties, and their moduli spaces.

...Sure you can define algebraic stacks without first defining any intermediate geometric objects. However, once this is done, there you are, and there is nothing that you can hold onto and relate the objects to… "

There is a book project in progress by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch. It has not been completed yet and it is not clear when it will be, but I find the existing chapters quite useful.

http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1