Medway has grammar schools.
Being single, I can't really say what it's like to bring up a family here. But there is a lot of open space. Children play out a lot. There are several family-type restaurants in the High Street. There don't seem to be many primary schools - but I suppose that rather depends exactly where you live.
Parts of Rochester are in a public transport black hole, so if access to bus routes is important you'd need to check that out. Not great for doing a big shop, although there are a couple of small supermarkets and corner shops. The whole Medway area isn't great for shopping, although Bluewater isn't far, or Canterbury.
Not good if you don't like hills, either!
The High Street in the past has had a bit of a reputation late at night, but I frequently get off a train late in the evening and walk home and have never felt unsafe. There are usually plenty of other people around.
Rail links to London and the coast are one of the reasons I moved here, and they're even better now. Good access to M2 and M20 too.