I'd try the British Surfing Association on Fistral beach as they're good with kids, have all been vetted / trained in kiddie conduct and safety is really tight.
They can be contacted here:
www.nationalsurfingcentre.com
E-Mail:
info@nationalsurfingcentre.com
Lessons can be quite expensive (with any school) so might be best to contact them 1st and find out how much you'll need to save!! I don't know if there's a kiddies rate.
You will want to be with a good school with tight safety wherever you go as the currents around Newquay are very strong in surf conditions (as they are in on any surf beach).
For general swimming in Newquay, Towan beach by the harbour is probably the beach most family orientated (you'll find a lot of families with smaller kids here) and has the mildest currents. You should only swim between the red and yellow flags on any of the beaches as these are the demarked safety zones which the lifeguards mark out as having milder currents. Small kiddies should be accompanied in the water (even shallow) at all times. You can't surf on Towan although bodyboarding is allowed. Bodyboarding is quite kiddie friendly also.
As for where to stay, I'm not sure for families as I never really visited Newquay whilst I was a kid, but be aware that central Newquay is very rowdy at night with large groups of drunken lads, so you might be best further out. There are quite a few hotels on Mount Wise and also guest houses on Tower Road which are a little further out. However car parking is limited in the centre and the town is very hilly so walking with small kiddies equally might be hard work. Tower Road and Mount Wise are within walking distance of the centre but like I say it is hilly.
Maybe someone else can offer good advice on family friendly accomodation.
I don't in any way want to put you off with what I've said. Newquay is a great place for families in the day, you just might prefer to be a bit further out at night.
Al.