I live in Falmouth and have two young DSs. I recommend the beaches obviously. Castle Beach is best for very young children as it has rock pools all the way along it which prevent very young children from running off in to the sea. It also has a steep slope back up to the road - also not easy for small children to escape up. Gylly Beach is a good all round beach, with a bouncy castle. Swanpool is good too, it also usually has a bouncy castle. Then you have Maenporth, which is more sheltered and better for watersports.
The National Maritime Museum is great for kids on a wet day, my two love it. Also, if it's raining, there is Raze the Roof in Penryn. An indoor soft play area.
Then there's Pendennis Castle, or if that's too expensive, walk around the moat and then walk down to Little Dennis from the car park on the point, and that's a mini castle which is free.
Getting a ferry to St Mawes or Flushing is fun for the kids, and/or you could take them on the King Harry Ferry over to St Mawes.
Falmouth Art Gallery is small but interactive, and has art sessns for kids.
Falmouth Aquarium has just opened up. We haven't been yet, but it looks good and is in the middle of town, so could be combined with a bit of shopping.
Argal Reservoir is good. It's a small charge to park, but there's a playpark and picnic area and the walk around it is around an hour and a half.
Then there are all the parks and gardens.... Phew! You don't need to go far to have fun in Falmouth, but if you have got a car it would make life easier and you could also visit some of the other places mentioned by other posters.