I'm from Bucharest (Romania) – I lived in the US for a while when I was a kid, and moved to London over a decade ago.
I've always lived in fairly central areas, so that's what I looked for when I moved to London. It makes most commutes much easier, and most bars and restaurants are just a short walk, cycle or cab ride away.
I (also) landed in Belgravia initially, where I spent about 4 years. It's as central as it gets, when it comes to predominantly residential areas. And aesthetics aside (those grand white stucco houses are always a sight for sore eyes), it's literally around the corner from Hyde Park and Green Park. From a neighbourhood perspective, though, you're not going to find the most engaged local community (there are a lot of embassies and official residences around) – but you do have some great pubs to hang around (see The Alfred Tennyson or The Thomas Cubitt).
I then moved to Notting Hill. It's also very central and residential, but with a much better sense of community. Its main hotspots (Portobello Road and Westbourne Grove) will be buzzing on weekends, but otherwise things are usually calm and quiet. It's got great transport links, with the Central Line covering its southern border and the Circle and Hammersmith & City Lines snaking along the northern side. You'll have everything you need within short walking distance, including great pubs (The Westbourne or The Oak), restaurants (Granger, Gold, Electric Diner) or cinemas (Electric Cinema). The one thing I would personally recommend, when looking for a flat around here, is to consider something that isn't in the middle of the madness (and definitely not on the Notting Hill Carnival route) – great examples include Sunderland Terrace, Northumberland Place, Chepstow Villas or Hereford Road.