I love Weymouth...stayed here many a time once
http://www.eastfleet.co.uk/
About a mile or three out of town, down a hill and onto the 'farm'. Good facilities, licenced, food if you want it, shop, good shower / toilet facilities last time i was there, quite busy (a bit of a negative, but a good sign) and it borders Chesil Beach.
Downside...Chesil beach (the bit the site borders) is not visually pleasing, but it is wild with nice walks either way.
Weymouth itself ?...i love it. Good mackerel fishing from the harbour / pier, fantastic long beach, quite a nice town centre, good array of shops, the Swannery lakes are nearby, always very relaxed there.
Portland is spectacular to go up, very rugged, beaut views from the top, you can see Weymouth and Chesil beach dissapear into the distance, Verne prison cut into the rocks....etc etc etc.
Rarely had a problem finding parking. Its busy, but once you know where all the car parks are, there's always a space to be found, specially by the harbour.
If it were'nt obvious....i love it
I should say...there may be better places in Weymouth, but i've no complaints about this one...