What The Lion said - either Staines (TW18 postcode) is nice for families, or for a very easy drive to Osterley, Twickenham/Whitton (postcode TW2) is great value, as it's the less fashionable end of Twickenham. The TW1 postcode is trendier and less suburban feeling but it's much more expensive than TW2. But both postcodes are in Richmond borough which is a very nice borough and the transport links are great. (As you've found, TW9 & TW10 are really pricey, but by moving to the edge of the borough you still get a lot of advantages of the leafy plushness without the stinging pricetag).
Reading - urgh, honestly, avoid. The really nice areas are expensive - certaintly more pricey that Staines or Twickenham/Whitton and the cheaper areas are just horrible (grimy student digs etc). The one way system is a horrific nightmare and the drive to Osterley would be a pain in the bum.
Anyway, do pop onto Rightmove and do have a look at TW2, TW16, TW18, TW20. Hampton is TW12 and it's lovely but it's slightly more pricey than TW2. TW11 is Teddington and you will get 20% less sq feet than in TW2, that's how big the 'value for money' difference is, so it's your call. As Kirsty says, space or location?
Avoid TW3, TW4, TW5, TW6, TW7, TW13, they will look like good value but for various reasons they may not suit.
Agree that renting for a short time (6 months) is a great idea first but bear in mind that after 4 years of almost static house prices in the abovementioned areas, things are starting to pick up noticeably and I'm startled at seeing house (selling) prices increasing strongly in the past 9 months. As a final (!) point, renting in the abovementioned postcodes is lousy value for money - rentals are much in demand which is pushing prices ever higher for renting - so the quicker you're out of rented and into your own place the better. Good luck!