Broughton Street, Leith Walk, Hillside or Bellevue. They are all pretty close to one another.
All are close to Waverley and also close to the last stop of the tram line that goes to the airport. All also very close to the centre of everything. You're also close to Calton Hill and the Botanic Gardens for green spaces, as well as lots of little green bits dotted about.
Broughton Street is very vibrant- coffee shops, bakeries, independent shops, restaurants, florists etc. Very urban, but most of the tenements have big back gardens. Great health food shop- RealFoods- at the top of the road. Has a fishmonger and a good butcher (Crombies) too. As well as Vino and Villeneuve Wines.
Leith Walk is similar and very up and coming. There are still little local shops, like Tattie Shaws the greengrocer, and that Edinburgh institution, Valvona and Crolla. La Favorita is a great pizza place. Plus there is a library on MacDonald Road. There is a Sainsbury's, a Co-op, a Scotmid (Edinburgh version of Co-op), a Tesco as well as the independent shops.
Hillside is full of huge Georgian and Victorian flats at not too dear for what they are. Easy to get out to Portobello or up to Calton Hill. Not so many wee shops as the first two areas I mention but still quite a few of those if you head out the London Road a wee bit.
Bellevue is a wee bit further from the station than the other three- think 15-20 minute walk rather than 10 minutes. But there are quite a few houses in that bit. Very close to the Botanic Gardens, also to a lovely food shop and cafe called Earthy. There's a pretty good Tesco on Broughton Road, and other local shops include a gluten free bakery (Sugar Daddies), the Marshmallow Lady, the Beerhive (independent off licence) and Union Yoga. Lots of little cafes too.
All those areas are pretty urban- but if you're coming from London they'll feel more open/less crowded without you feeling like you've gone to the ends of the Earth either.