I live in Chichester and can vouch for going there, though lots of what I would usually recommend (the cathedral, the pallant house art gallery) will bore your age children! Worth going, though, even if for the pretty market cross in the centre of town. Priory park, just off of North street, is spacious enough for a run around if they get bored and has a nice children's play area with things to suit the toddlers and older ones.
If you want stunning views, head for Goodwood and the top of the trundle where you can see the whole of Sussex and beyond spread out beneath you.
If you want a family day out away from Butlins, mine always enjoyed the Weald and Downland open air museum. At the age of your oldest mine were fascinated by the reconstructed old houses there and how people used to live, and it's not stuffy or stuck up at all - you can explore to your heart's content and there's loads of run around space to stop your 5 year old getting bored. They usually do activity days once a week in the summer holidays.
The Fishbourne Roman Palace would probably appeal to you eldest too - often in the summer they also have days where people dress as soldiers or eat Roman food, or have battles.
Bognor is a shite hole. If you walk west along towards Felpham, that's much nicer and a prettier place with some attractive beach cafes. Littlehampton, a bit further still, has a promenade train and much more in the way of play parks, activities, and beach shops to appeal to your age children.
I can't tell you much about inside Butlins, but for outside, feel free to ask!