For a long time I have been looking for a sensible introduction to p-adic Hodge theory, and I think I might finally have found one: these lecture notes of Conrad and Brinon, an expanded but still prelimary set of notes based on their CMI summer school lectures earlier this year. Thanks to David Brown for pointing out these notes on Math Overflow, as part of an answer to a question about models.

A much shorter survey is Berger: An introduction to the theory of p-adic representations, but Conrad and Brinon give a lot more background, which seems very helpful.