Hi Jo-Anne. All the museums (apart from Commonwealth Museum) in Bristol are free. The City museum at the top of Park Street is a good place to start - they've had a good stab at covering the Neolithic.

There is a good, scenic, hillfort (iron age) in Liegh woods - have a look at web entries for the area on this site.
Like Dee says, Stanton Drew is just under 10 miles out of Bristol - it's a great henge, and never busy, while Sony Littleton is one of the nicest longbarrows that I know of.