I take 4 year old DD to the Hornimans fairly often, it's a fantastic day out, lots of things to see, lots of interaction for little ones, and an aquarium which is all free.
They have a really good programme of activities for the summer holidays including nature trails, storytelling by a theatre company with all sorts of props, puppets and actors and lots of singing and dancing. They also have arts and crafts activities (today was mask making) all of which is completely free. On top of that they have huge gardens with lots of space to run around and have picnics, and a little animal enclosure outside with goats, geese and peacocks.
I would highly recommend it, DD and I spent 5 hours there today and she was occupied and happy the entire time, despite the fact that it was raining and she couldn't play in the gardens!
Have a look here to see what's on.
P.s. I don't work for them or anything, I just think it's a great day out for the school holidays!