When I first arrived in the UK, living in Central London, I started flying with the Pilot Centre at Denham. It is (relatively)easy to get to as VP8 says - trains leave from Marylebone Station frequently and take about 35 minutes I think. They will indeed pick you up if you ask thme to, otherwise it is a 20 minute stroll to the airfield.
Denham School of Flying (Cabair) is on the easier side of the airfield and also very good, albeit more expensive by the hour. PFC actually has very competitve rates for hire/instruction.
Elstree is also pretty accessible I think, although I have never taken public transport there myself so I don't know the details.
Denham is a nice place to fly from although (as I recall - no doubt someone will correct me if I am wrong) since it is inside the Heathrow zone with a designated access route, it requires 10k vis to take off/land as you effectively are operating under a SVFR clearance.