For an example of an application of $p$-adic Hodge theory in a geometric setting, I thoroughly recommend reading the beautiful paper

P. Berthelot, H. Esnault, K. Rulling, Rational points over finite fields for regular models of algebraic varieties of Hodge type $\geq 1$, Ann. of Math. (2) $\bf{176}$ 2012, no. 1, 413-508.

The result is nice and concrete: they prove congruences for the number of rational points on such varieties. As you will see, the proof makes use of a nice range of big theorems from $p$-adic Hodge theory and is very clearly explained.