You have chosen a fantastic place to go with small children. A trip on the St Ives Bay branch line is a MUST! It is a beautiful journey and the children will love it! I'd recommend using the branch line to go to Carbis Bay, and making the most of the fantastic beach there. Carbis Bay beach does not get anywhere near as busy as the beaches in St Ives, the sand is soft and the ocean (usually) calm and clear. Toilet facilites and shop / cafe available.
Days out would definately include Paradise Park at Hayle. If you go on a hot day, go in the morning when everyone else is at the beach, and you will be in with a chance of feeding the penguins (and stroking them / photos after if children would like!). Make sure you check out the times for feeding the farm animals too. Paradise Park is also good on a wet day as they have a soft play centre attached to it. Take a picnic and stay all day! Hayle is only 5 minute drive from St Ives (traffic permitting).
I'd also recommend going to the Seal Sanctuary at Gweek (about 45 mins) away. I think your children are a bit too young for Flambards - save it for another time.
Mousehole is a very pretty place. Make sure you get the book 'Mousehole Cat' for the children. If you buy it beforehand you can go on search of it!
Lands End is also good - although you will need to wrap up warm as it does get windy! There are things for the little ones to do there. You pay to park and only pay for the activities you want . There is a play park there with lifeboats etc. that are free, take a picnic and it could be a cheap day/half day out. Sennen Cove is a lovely - you could spend half a day on the beach there and then go to Lands End (very close).
St Michael's Mount is also fantastic, getting the boat to the island is a great experience, but there are a lot of steps to climb when you get there so depends how your children will cope with them.
When in St Ives (especially around the quayside) look out for the seals - they usually show their faces when the fisherman are coming home!
Also go to Gwithian and walk along the cliff tops - there are usually families of seals to be seen.
Have ice creams in MooMaids of Zennor and eat inside. Beware of the seagulls - don't let the children eat outside unless they are under some kind of shelter (beach tent, brolly, overhead wires all ok) - the seagulls swoop down and steal food ... and don't let the children feed them their chips - it only encourages them to steal more food later on .... and they can be vicious!
There is loads of stuff for little ones to do .... a week won't be long enough!