My family and I have been going to Tilgate park for as long as I can remember, and still go every now and again. It is perfect for families with young children, older children, teenagers and their friends, couples... basically everyone! It's in Crawley and is pretty easy to find, it's free entry but you have to pay for parking.
Things you can do there include fishing, canoeing, arts & crafts, visit the animal park and Go Ape. Or you don't have to do anything and you can just take a walk around the lake. There is a Smith & Western for massive portions of spicy food (they also do Oreo milkshakes and an alcoholic Oreo drink!), and in the Walled Garden there is a little cafe and picnic area (this place does some of the best chips I've ever had).
I always forget how huge this place is and how much there is to do, 'cos my family tend to stick to the same tradition every time we come here. We walk halfway around the lake and get to the walled garden where we have chips and tea at the cafe. Usually after our tea and chips we either walk back the way we came or walk around the rest of the lake and back to the car park. It's a lovely, scenic place to just get out for a stroll.
Here is the link to the Crawley council page about Tilgate if you want to find out how to get there.
And here are some photos from my last visit. I've never been when it's been this hot before and the lake never looked more stunning :)