I'm based in Brighton - this is a decent site with lots of suggestions which we used to find stuff to do -
https://www.dayoutwiththekids.co.uk/things-to-do/south-east-and-london/east-sussex/brighton - it's got lots of stuff, but much of it is smaller and more local to Brighton so perhaps not full days out, but an afternoon's entertainment.
For the bigger stuff, as well as Chessington (and Thorpe park which was mentioned) there is also Legoland which is about 1.5hrs by car - traffic on the A/M23 can be unpredictable though, especially on the weekends
The mainline train into London is only 50 mins which makes everything there very accessible e.g London Zoo, Planetarium etc. etc.
There are quite a few smaller, privately run wildlife/farm/amusement parks, but they tend to have limited appeal to older kids (I'm thinking your 12 year old). but for them there is go-karting, the indoor one in Lancing is great - our kids love sky high trampolining in Peacehaven (10 mins outside Brighton by car - they are knackered after an hour but you need to book), GoApe in Crawley (30 mins) and there are several pools if the sea isn't your thing (the Triangle in Burgess Hill has pipes/flumes 20 mins).
There are also a large number of National Trust gardens dotted around the local countryside to walk and picnic in if that's your thing. Nymans is great for a tramp in the woods but like most places gets very busy if the weather is good.
More local entertainment in the town itself (apart from the standard pier funfair rides and arcade) include a many great places to wander and eat, especially visit the North Laine which has more independent (non-chain) shops and places to wander/eat/sit/drink... if food is your thing there is lots of choice, highlights include Cin Cin (Italian -
http://www.cincin.co.uk), La Chosa (Mexican -
http://www.lachoza.co.uk), Oshio (Japanese -
http://oshiobrighton.com) and StreetDiner every friday lunchtime is a great street food market with lots of choice.