Hiya, I lived in the Maidstone area until a couple of years ago, and still have family and friends there. Coxheath is nice, so is Boughton Monchelsea, good local primaries, quite small and nice village community feel. With secondary schools some are v v good and have become oversubscribed like Valley Park, but that is on the other side of town nearer towards Downswood that you mentioned. Maidstone is unusual having 4 Grammar schools as well. If you did live in Coxheath and didn't go to one of the Grammars, you'd probably get Cornwallis Academy, (Ofsted Good) or depending on distance from your home maybe New Line Learning Academy which is currently Requires Improvement. Saying that, you don't need to think as far as secondary schools yet! Congratulations by the way 
There's a lovely area with Outstanding primary schools that a lot of people with young kids move to for this reason, but house prices reflect this and are £££, have a look at Bearsted. If that's out of your budget there is also Madginford which is very close by (pretty much between Bearsted village and Downswood), where your budget could get you this:
www.rightmove.co.uk/property-for-sale/property-80699570.html
There's also Grove Green up the road, with another outstanding primary. Vinters Park is nice and houses cheaper than Grove Green. There's a lot of new house building going on in Otham and I know a few people who live there, it's a bit further away from the town but still only 15 minute drive, I think it's called Langley Park and they've built shops and a primary school to accommodate for all the new houses.
I also know of quite a few families who have moved to Ashford where there's been a lot of new housing development, because it's cheaper than many parts of Maidstone. It's about 20-25 mins down the M20 from Maidstone.
Feel free to pm me if you have any specific questions, I lived in the Maidstone area for many years!