Family friendly areas Stockholm & Rental agents

Hello, we are relocating with our 2 daugthers from 9 an 13 to Stockholm this coming summer, and are already in the process of looking on the internet which areas would be the best to live in. Most people I have spoken with recommend Danderyd and Taby, but maybe you can also advise us on good areas...we have a maximum of Protected content to spend a month in rent.
It has to be close enough to city life, but a little bit outside with public transport and shops around would be perfect.
Also, can you recommend good estate agents that don't ask the world for mediation, as it seems big business here in Stockholm to charge enormous amounts for that...