Putney is very well connected as you will have the choice of East Putney on the district line ( good for West London)and Putney rail (for Clapham junction, the City and Kew and Richmond and beyond). However I did live within 10 minutes walk from both of those and from where you will be, it would be more like 15-20 mins.
But there are lovely pubs along and by the river and riverside is good for a walk/ run or cycle, Valentine's- an Italian deli and restaurant and loads of new places I'm sure. There is the Putney school of Art nearby if you want to take up a course. It feels cosmoplitan with Fulham across the river. Southside will be near as well. The parks are all gorgeous and your nearest one would be Wandsworth park. I lived there between 2010-14 but have heard from friends who are still there it has got a lot busier since I left because of the new apartments suddenky springing up along the upper Richmond rd and the pressure particularly being felt by families looking for primary schools. Brandlehow' s catchment has shrunk even more.