Farmer Palmer's is good but not what I would call a "cheap" day out. Would still cost over £50 for admission but there are parts that are covered so good for rain as you can go indoor/outdoor depending on the weather.
Flip out has just opened in Poole, similar price though but would be well suited to your kids ages.
Poole Park is nice. Few outdoor play areas, miniature railway (not sure when open though), mini golf, lake to walk round, cafe and soft play.
Upton country park good, all outdoor though so need coats. Cheap.
Mr Mulligan's crazy golf. Bit expensive but a cool experience all indoors, they also have ping pong and darts. (Book in advance it gets busy).
I personally wouldn't recommend oceanarium, it's expensive for what would be 45 minute experience (although it is an all day pass).
You didn't ask for food recommendations but would highly recommend Chez Fred for great fish and chips.
If I think of anything else I'll add!