If you only want to change twice (at Reading and Oxford), the cheapest option is to buy a GroupSave discounted Off-Peak Day Single, costing £86 in total for the four of you.
If you're prepared to change additionally at Redhill, this opens up a slightly cheaper option involving splitting at Horley and Blackwater, provided you're departing Gatwick after 12pm or so*. This would cost a total of £79.80 for the four of you, again with GroupSave discounted singles.
It's possible that cheaper Advance tickets might become available nearer the date, but if you are coming off a flight you might not want to buy one of these, lest your flight is delayed. That being said, this doesn't have to be a problem - I typically buy Advances for around 1-1.5 hours after my flight is due to land, from a retailer that allows me to change my ticket without a fee. If my flight is starting to get delayed and I'm at risk of missing my booked train, I change the ticket to a walk-up ticket (or Advance on a later train), just paying the difference. This way at worst I pay the walk-up fare, and at best I pay less for an Advance.
*The ticket between Horley and Blackwater is an "Evening Out Single" for which you'll only obtain itineraries to depart Redhill from 12pm onwards; the first train from Redhill after 12pm which calls at Blackwater is the 12:43.