Explanation of 1-generic to prove undecidability of halting problem
A finite prefix of  $G$ is $G \cap \{0,1,\ldots n\}$ for some $n$. It is easier if you look at the characteristic function of $G$ which can we viewed as an infinite word in $2^\omega$. Then a finite prefix is a finite initial part of $G$.

For forcing, there are many resources, depends on what you want to learn. A point to start is the Wikipedia pages and the references listed there: Forcing (set theory), Forcing (recursion theory). Timothy Chow's article A Beginner's Guide to Forcing is a good introduction.