Can't forget the Post Office Vaults, with its catalogue (literally) of bottled world beers. Pricey but awesome.
These are all good suggestions for central pubs. I would add The Victoria by New St Station which I assume is still fun and serves proper Hendricks G&Ts. I don't get out much anymore.
If you're just staying round the town centre I'd recommend the Old Joint Stock, The Old Contemptibles, any pub around Charles Street. You've also got Bacchus Bar off New Street, plus there is a Brewdog bar as well serving London priced beer that's strong as fuck, but it's full of hipster douchebags. If you can manage it try the Lord Clifden in the Jewellery Quarter which is a pretty decent pub and they do iced Thatchers Gold Cider on draught which I can't get enough of. It's a like goddamn alcoholic slush puppy!