From which sources would you learn about crystalline cohomology and the deRhamWitt complex?
With enough enthusiasm, I would try to learn about crystalline cohomology and the deRhamWitt complex from the homonymous article by Illusie:
Illusie, Luc. Complexe de deRhamWitt et cohomologie cristalline. (French) Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501661. MR0565469 (82d:14013)
Fortunately, it is publicly available at:
But this is most usefully read as needed after one is acquainted with the following also relevant references (perhaps in this order):
S. Bloch, Algebraic Ktheory and crystalline cohomology, Publ. Math. Inst. Hautes Etudes Sci. 47 (1977), 187–268.
O. Hyodo and K. Kato, Semistable reduction and crystalline cohomology with logarithmic poles, in Periodes padiques (BuressurYvette, 1988), Asterisque 223 (1994), 221–268.
O. Hyodo, On the de Rham–Witt complex attached to a semistable family, Compositio Math. 78 (1991), 241–260.
O. Hyodo, A cohomological construction of Swan representations over the Witt ring. I, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 300–303.
If you're just looking for a quick overview, you may want to read the lecture notes from the 2009 MIT Ktheory lunch seminar, especially the first five lectures
If your French is good, I would suggest a recentish survey of A. ChambertLoir (http://perso.univrennes1.fr/antoine.chambertloir/publications/papers/cristal.ps.gz, published in Expos. Math. 1998), which is based on lectures by L. Illusie, A. Mokrane and him (from 1995).

2I presume the following would be a useful reference, had I had a chance to look at it... (By the way, could anyone provide an electronic version?) Fontaine, J.M.; Illusie, L. $p$adic periods: a survey. Proceedings of the IndoFrench Conference on Geometry (Bombay, 1989), 5793, Hindustan Book Agency, Delhi, 1993. MR1274494 (95e:14013) – monodromy Feb 26 '11 at 21:17

Indeed this paper indeed seems a good read, but I was not able to find it anyhow. What about the electronic version ? I'm interested... – Matthieu Romagny May 13 '13 at 19:25

You may find it helpful to read Illusie's survey article in the proceedings of the 1974 Arcata conference "Alegebraic Geometry" published by the AMS. It's quite a nice account of the basics.

3
Lots of good suggestions already, but here are some references:
1) Illusie's articles (Arcata 1974, Motives 1994, Lectures Notes in Math 1016)
and IllusieRaynaud (Publications IHES)
The third article is a masterful summary of the main results of the de RhamWitt complex.
2) MazurMessing Lecture Notes (excellent introduction to onedimensional crystalline cohomology) and Mazur's article in the Bulletin of the AMS (Hodge filtration??)
3) Michel Demazure's Lecture Notes on pdivisible groups (Dieudonne modules)
4) Nick Katz's articles Slope filtration of Fcrystals and "Crystalline cohomology, Dieudonne modules, Jacobi sums) (some conference proceedings) with a concrete application of IllusieRaynaud's results to arithmetic.
5) Kedlaya's survey article (2005/6?)
If you don't mind reading mimeographed notes, the exposition by Berthelot and Ogus is certainly worth taking a look at. It has the added advantage of only assuming a minimal background in algebra and algebraic geometry. If you find the DJVU version through Google, every other page is slightly cut off at the right, but this did not interfere with my reading of the document.
After you learn the subject, perhaps if you're feeling ambitious enough to learn some padic Hodge theory from Gabber and Ramero's long treatise (admittedly only lengthy because it starts from fundamentals), you could tell me if it treats the crystalline topoiI have only skimmed it and it seems to only consider Zariski and etale sites.
Good luck. (Addendum to Pete Clark: Feeling a little jealous here ;)