I'm going to info dump sorry but I have lived here for 10+ years so I think I'm qualified to say that this is a good list for avoiding high traffic areas and to see more of what actual locals do (mostly south london because it's where I live)
Eat:
-Kudu Grill (Nunhead) also there's a Little Kudu in peckham although I've yet to eat there so cannot say whether its as good
-Theo's Pizza (Camberwell / Elephant & Castle)
-The Camberwell Arms (Camberwell)
-Artusi (Peckham)
-Brutto (Farringdon)
-Max's Sandwich shop (Finsbury Park)
-Jai Krishna (Finsbury Park) (rogue recommendation but you have to try a good Indian when you're in London I'd say)
-Casse-Croute (London Bridge)
-Manze's Pie & Mash (Peckham)(try the jellied eels if you're feeling brave)
-Speedboat Bar (Soho)
Books:
-BOOKS PECKHAM (my fav 2nd hand bookstore and the owner is nice)
-Bookmongers (Brixton)
-Tenderbooks (Near Leicester Square station)(really cooool art books, the whole street it's on is known for having lots of book shops, worth a visit and very central so it's easy to find!)
Pubs (London essential, sorry about the pint prices though):
-Skehan's (Nunhead)
-Faltering Fullback (Finsbury Park)(try find a seat in the beer garden if poss)
-The Camberwell Arms (Camberwell)
-The Grapes (East London-ish)(fun fact: it's co-owned by Ian McKellan of Gandalf fame)
Also the parks are great to visit in summer:
-Kew Gardens (the general area is nice to be in - lots of expensive homes and english quaint-ness)
-Hampstead Heath (it's is a good place to walk around in for a day, the 'village' part is cute too like with Richmond/ Kew Gardens, lots to do here)
-Crystal Palace Park (for the cursed dinosaur sculptures mainly)
-Greenwich Park (a tourist classic but I've included it because it's just a nice park and a good alternative to Hyde Park)
-Highgate Cemetery (technically not a park but you can visit Karl Marx's grave which is cool!)