We've "lived" in several areas around London in the past few years by exchanging houses for 2-4 week periods. We made sure all of the locations had easy access by public transportation into central London.
We stayed in Twickenham, which is near Richmond and Chiswick. We liked all 3 of these cities a lot for their proximity to the Thames, tube/train connections, proximity to attractions to the west like Kew Gardens, Hampton Court Palace and Windsor. Richmond and Chiswick especially had very charming high streets. With the convenient transportation connections we could be "home" in about 30 minutes after a west end show.
A few years earlier we stayed outside of the city in St. Albans because we wanted to drive to the north and explore a bit more. This turned out great because St. Albans had so many historical layers of its own and a great rail connection (Thames Link) into London that zipped us to the center with very few stops in about 20 minutes.
This year we are exploring another home base, Ealing, which has a train line and 3 tube lines. They also have lots of busses which come about every 5 minutes for other travel options and directions. We are finding it to be very easy to get to and from and are enjoying our time also exploring places close to us when we aren't sightseeing in London for the day.
If you want to get the feel of living in a neighborhood and observing daily life in a charming neighborhood I'd encourage you to rent a flat after carefully checking on actual travel times between your home base and where you want to sightsee. Two apps, rome2rio and citymapper really help!