Hi - I work in London every week, and used to live here, these are my recommendations.
Pubs (craft beer):
Mother Kellys (
http://motherkellys.co.uk/) - Short trip east on the Central line. Always excellent modern British and European craft beer, all keg, no cask. Also an amazing bottle selection.
The Euston Tap - Tiny pub outside Euston Station (very central, easy to reach). Loads of British craft beer (cask, keg + bottle), where beer nerds rub shoulders with commuters who just want lager and don't understand why there's no Heineken. Upstairs lounge is very sociable. I've had a lot of very messy nights up there. The tables get sticky and occasionally the whole pub smells like a toilet. This is part of the Euston Tap magic, don't question it.
The Dovetail, Clerkenwell (
http://dovepubs.com/aboutdovetail/) - Close to Farringdon tube station (Circle line) or about a 15 minute walk from Kings Cross. Basically a Belgian beer cafe, but in London - loads and loads of Belgian beers on tap and in bottles, and great Belgian food, with a nice authentic cosy European vibe inside. Clerkenwell is a really nice historic part of London, formerly home to Charles Dickens and Karl Marx.
The Craft Beer Co (various pubs in London) - can be expensive compared to other pubs, but the draft selection tends to be huge and they often have amazing things like Cantillon straight lambic on tap.
Brewdog (various pubs in London, by far the best one being in Shepherds Bush) - a bit soulless, popular with yuppies, but always reliably great beer (keg + bottle only). They also have a bottle shop near Kings Cross, called BottleDog, which is great, and has its own little secret bar in the basement.
Pubs ("Traditional"):
AVOID any pub run by Greene King or which serves Greene King beer. Their beer is terrible and a national laughing stock. Do you enjoy cask "IPA" at 3.7% ABV that tastes of defrosted freezer moisture? No? Then stay away.
The Clachan (close to Oxford Circus tube station) Part of the Nicholson's chain, you find a good range of cask beer in these places and it's usually well kept. The Clachan is a very useful pub to hide in, because it's in the middle of central London's main shopping district.
The Viaduct (as mentioned above) - this is opposite my office! It's part of the Fuller's Brewery pub estate, they have dozens of pubs across London, and they vary quite a bit. The beer here is well looked after but the selection is uninspiring - London Pride, Sierra Nevada pale, Chimay Gold, macro lager. Worth a visit if you're in the area (it's an interesting building) but you would be crazy to make a journey just to visit this pub.
The Parcel Yard - this is Fullers "flagship" pub and it's actually located inside Kings Cross railway station. Loads of different beers, the entire Fullers range and it's probably the only pub in London where you can pick up bottles of Fullers Vintage Ale (their annually released barrel-aged barleywine) which are 10 years old or more. Very good food, really nice historic interior, very spacious. Gets busy with commuters between 5pm - 6:30pm.
The Queens Head (
http://queensheadlondon.com/) - 5 minute walk from Kings Cross. Run by same people as Mother Kellys. Looks and feels like a classic (some would say "idealized") London pub, but serves really good craft beer (cask and keg).
Breweries
The Kernel - unfortunately they closed their tap room due to rowdiness but you can still visit at the weekend to purchase bottles. They make some of the best beer in London, they pretty much introduced the classic "US-05 IPA" style to this country, and they do that style extremely well alongside their historic recreations of London Export Stouts, and some really good sours. You see their beers in various pubs in London and they're always really really good.
Brew By Numbers - close to the Kernel, they have a great taproom although I haven't been there, my friends say it's great. Probably the best brewery in London right now. Their hoppy beers are total juice bombs, they make great saisons, and they do cool stuff like white wine barrel aged tripels.
Fullers - a bit of a hassle to get to via public transport, but if you want to visit a historic British industrial brewery without leaving London, this is the one.
I hope this helps!