I performed LME with fixed and random effect. I can see the p value for fixed effect but I don't know if I have to report p-value for random effects. And how to get this p-values (if possible).
What hypothesis would you be testing? That something like the variance is zero (or some other value), or that some of the condition modes differ (this is a common question in some educational context and Goldstein and Leckie discuss this ... and Goldstein in several earlier papers too).
Section 1.6 of http://lme4.r-forge.r-project.org/lMMwR/lrgprt.pdf seems relevant.
I believe in the lme4 paper in JSS (or somewhere) Bates et al talk about the importance for the computations that zero variance can be given (and lots of packages will give negative variance values too). This requires careful consideration of the hypothesis. As far as the fixed effect p values, it is worth linking to http://127.0.0.1:21752/library/lme4/html/pvalues.html
I have one model with fixed effects and random effects. It is easy to get the p-value for fixed effects, but I don´t know how to get the p-value for random effects (participants and items).