Stay put, we're already flooded with people fleeing London.
Seriously, there are some good points made here and some not-so-good points.
On education, depends what you're looking for. If you want grammar schools, don't go to Sevenoaks because there aren't any. I can't tell you much about the schools in Sevenoaks, since the only one I know is Sevenoaks School (the public school), and I don't have fond memories.
IMO Tonbridge's schools are better than Tunbridge Wells. Tunbridge Wells is pretty good, but it doesn't have the range of schools of Tonbridge.
Tonbridge's main industry is education - something like 9,000 schoolkids go through the town every day to one of the half-dozen grammars, or the 6th form college, or the tech college, or Tonbridge school, which is one of the best public schools in the country.
Bear in mind that entry to grammars is through the 11+.
For primary age children there's a good range of schools but you have to be really careful where you buy because catchment areas are tricksy. Check the details on the house to see which catchment area you're in. The best one in Tonbridge is supposed to be Woodlands. There are also plenty of private primary (prep) schools, most of which offer entrance to the public schools as well as 11+ training.
On connections to London, Sevenoaks is your best bet. 25ish minutes to London Bridge. Something like 1 in 4 of the people who live in Sevenoaks travel up to London every day. Tonbridge is second best, since it has two main lines (Ashford/Canterbury and Hastings) running through it, about 35 mins to CharingX. Tunbridge Wells is the worst of the three, about 50 mins train journey.
On house prices, Tonbridge is your best bet. You'll pay half what you would in Sevenoaks for an identical house. Tonbridge isn't rough or unpleasant, but it isn't a place you'd want to shop apart from the Waitrose and a couple of newer shops (nice coffee place, new butchers, couple of nice restaurants and gastropubs). You'll still be paying £250k for a 3-bed 60s semi, though.
For sheer up-your-own-arseholeness, Tunbridge Wells takes the cake and eats it all. 
You will need a car. This isn't London.