Bewilderwood involves lots of climbing up and down ladders, walking along platforms suspended in trees etc, as well as a great den building area. Could easily spend all day - but have heard other people comment that there is not enough to do...
Also Out of Africa (large zoo type place), at Kessingland just south of Lowestoft is a good day out.
As catherine says, Southwold is lovely, beach is quite stony at the seashore so useful to have beachy type shoes that you can wear in the water. Went there last week, and was gorgeous - hot but windy. DS1(13 yrs)and DS2 (9) really enjoy crabbing, which you can do by the river near to where you get the ferry to Walberswick. Cannons on Cannon hill in Southwold are always a novelty to look at/climb on/have your photo taken beside. Southwold has a pier with a few shops, also lovely shops in High Street - lots of independent ones (though Fat Face has just appeared on the scene). Fewer shops at Walberswick, as I recall. Lowestoft is more of a "resort" type beach, not sure what the actual beach is like. Kessingland beach is supposed to be good but have not been there.
Fritton Lake (on A143 towards Gt Yarmough) has big adventure playground where you can just laze and keep an eye on children as they bounce around...also (as name suggests) lake on which you can hire boats.
Will carry on thinking, hope this is useful...
Day trip to Norwich would be do-able from Bungay I think if weather bad.