Hi OP - the most "pleasant to live in" parts of London are -
Chelsea
South Kensington
Knightsbridge
Holland Park
Notting Hill
The above are 'prime locations' though with prices to match.
A little further south-west (but still in zones 2-3) you have -
Parson's Green
Fulham
Putney
Barnes
The above are fantastic for families because if the schools, relative safety, access to the Thames, gorgeous river walks and loads of green spaces. Still expensive, but not as much as Chelsea etc.
In West London, you have -
Chiswick
Ealing (inc parts of Acton)
Again, very family- friendly with everything you can think of in terms of shops, restaurants & facilities. Feels a slightly more "busy" perhaps than Barnes etc as the river is less visible and roads are going out west to the A4, M4.
NW London there is -
Hampstead
Highgate
Expensive again (but not as much as Chelsea, Knightsbridge etc). Loads of character, "villagey" atmosphere - everything you could want inc. open spaces (Hampstead Heath). No River Thames here though.
Yhode are probably the most sought-after places to live for families.
East London (has many very cool and trendy areas too, which may or may not be your vibe.