Fulham....or other areas?

Any input on Fulham? 
We are moving to London in a couple of months and are trying to narrow down the areas to look once we get over there. 
My husband will be working near the Tower Hill Tube and we want an easy commute to there. 
I will be staying home with my 2 & 3 year old and don't want to have to drive.  Is Fulham suitable for a young family? 

Any advice on family areas (not Fulham) with a relatively easy commute to Tower Hill would also be greatly appreciated!
Sue

Wanstead!

The boy said that it's a good place to live...if it wasn't too far from where we have our social life then he'd be interested.  Lots of nice restaurants and bars for social life.  Good living quarters in the area.