Mr. Beard has given you a great list of specifics. If you want to get a sense of what the Anzio campaign was like in general, go down to the library and check out "Up Front" by Bill Maudlin. It is one of the best accounts of the war ever written from a grunts-eye view, and he was in Italy for a lot of it. His other books are good too, if you are close to a library big enough to have them.
Anzio was a place in Italy. The "beachhead" is the thin strip of beach that landing forces first occupy.