What sort of budget are you looking at for renting?
I would rule out christchurch as it's a bloody nightmare of a commute to Poole. Also the run from Bournemouth to Poole in rush hour will take over an hour. Look at surrounding areas of Poole such as Broadstone, Corfe Mullen, Wimborne.
I live in Broadstone, it's a small village type place, about 10 mins drive from Poole. Outstanding schools and two grammar schools close by as well. It's a very nice, quiet place here but loads to do in terms of activities for the kids, swimming pool, youth clubs.
Search Broadstone first school, Springdale first school and Broadstone Middle school which all of your children would go to in this area. Secondary schools start at 13 here and there is Corfe Hills and further out is Queen Elizabeth school in wimborne.
I am loathe to recommend any of the Poole schools as don't really know them too well but they are start at age 11.
Are you kids boys or girls? What sort of things do they like doing?
There is so much for them to do here, lots of water based activities, kayaking, swimming, sailing, surfing. We moved from Glasgow five years ago when my dc were 13 and 6 and despite their reservations, they have absolutely thrived here.