We go to Gorleston beach (just south of Great Yarmouth) quite a lot - mainly 'cos the Il's live there though! It's probably the quickest beach to get to from here, straight down the A134. It's a beautiful beach, was in the Guardian guide to family days out last week as one of the best swimming beaches in the country. It's not brash and over developed like Yarmouth, just a few car parks and one small beach cafe with huge queues!
We also go to Bressingham steam museum quite a lot but I think it's quite expensive for what it is, about £12 per adult and I'd say you could spend about 4 hours max there.
Diss (where I live) is pretty, lots of quirky independent shops, lovely big park based around a Mere, Gypsey horses to feed apples to on the common, small Farmer's market 2nd Saturday of the month, ordinary market every Friday.
Wortham Ling is quite nice for walks and picnics, no facilities like toilets or ice cream vans though.
The Eye Show is on August Bank Holiday if you're here then. Not an optometerist convention as I first thought when i moved here, but a County Show in the town of Eye. It's quite a big one and usually has lots to do.
I'm sure I'll think of lots of other things as well! Feel free to CAT me if you want directions/details etc.