To answer your first question last, connections to central London are amazing, via either trains to London Bridge or via the Overground. With both routes running through FH station, there are trains every few minutes.

There is a vibrant but small 'high street' comprising the end of London Road and Dartmouth Road as far as Forest Hill Library and the new pools. There are a few 'normal' shops - Sainsburys, WH Smiths, Boots, but there are many more independent shops, including one of the best butchers in London (The Butchery), retro sweet shop / ice cream parlour Sugar Mountain, award-winning card shop B cards, boutiques like Bunka and Marie's Boutique and many others.

Places to eat and drink (among others):
Canvas and Cream
The Teapot
St David Coffee Shop
Sylvan Post
The Montage
The Hill (formerly Question bar)
The Dartmouth Arms

Is it safe? Absolutely. You may read a few posts on here about mugging and burglaries, but remember that no one ever posts to say that yet another day has gone by when they haven't been a victim of a crime!

Have a look at this thread here, which asked similar questions and got lots of interesting answers:
http://www.se23.com/forum/showthread.php?tid=8659

Come and see for yourself - Forest Hill is a lovely place to live and you won't regret it.