We live quite near the site. Can't say I have visited it since they made it adults only. You probably need your car to visit most places as don't think there is a bus stop near the site.
Cockermouth is about 10 miles away. It is a lovely time with some lovely pubs and resturants. Carlisle is about 30 mins away. Depending on what you like there is Tulli House Museum always got something to see, Carlisle Castle and The Lanes which is a shopping Centre. Keswick is about 35 mins heart of the lake district so lots of lovely walks etc. Penrith is a bit furthe maybe about just under an hour.
Steve, You could struggle to fill a day in Maryport, unless there is something special on, they do have an International Blues Festival in the summer at some point, if you google it you should find the dates, about 20 minutes north of Maryport on the coast is Silloth, a victorian seaside resort, with nice beaches en route at Allonby.
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Steve I live in Maryport there is no way you will be able to fill a full day here. There is the aquarium and a roman museum and thats about it. Maybe half a day. You could maybe join it up with a trip to silloth.
Amazing family weekend with old steam engines, classic car displays, market stalls, and full catering and bar. And camping on site - Save £25 by booking in advance.