Can anyone suggest a good thorough reference for $p$-localization and $p$-completion of the integers? I'm an algebraic topologist who's found himself washed up without any intuition.

In particular, here are the two questions I'm trying to answer right now:

Question 1: what is $\mathbb{Q}/\mathbb{Z}_p$ tensored with itself?

I mean $\mathbb{Z}_p$ to be the $p$-local integers. The tensor is over $\mathbb{Z}_p$ or $\mathbb{Z}$ - the answer should be the same.

I know $\mathbb{Q}/\mathbb{Z}_p$ is an injective $\mathbb{Z}_p$-module, and $\mathbb{Z}_p$ is a nice local ring, so maybe something can be said about a flat resolution of $\mathbb{Q}/\mathbb{Z}_p$?

Question 2: What is known about the cokernel of the map $\mathbb{Z}_p \rightarrow \mathbb{\hat{Z}}_p$?

I mean the map that $p$-completes the $p$-local integers. I think the cokernel is a rational vector space, but of finite or infinite dimension? Does it have any nice description?