Proper Eastenders will probably tell you the East End isn't what it used to be, but it is still worth a visit both for historical reasons and recreational reasons - best thing is to go to Old Street tube station and walk down Old Street until it meets Shoreditch High Street, around that area there are lots of bars and cafes (for instance the Bricklayers Arms in Charlotte Road, Mother Bar on Old Street (which is free and open late), Jaguar Shoes and Catch on Shoreditch High Street and several cafes in Hoxton Square - also in Hoxton Square you'll find White Cube Gallery which sometimes, though not always has some interesting exhibitions, and it's free). There's a nice little record shop on Old Street which might be worth checking out but i think you're better off visiting the second hand record shops in Notting Hill Gate or the ones in Soho on Berwick Street or in Camden. Anyway, if you visit the (E2) area on a Sunday, make sure to see the Columbia Road flower market which is well worth a visit. On the other side of Cambridge Heath Road, around Roman Road/Globe Road there are two nice vegetarian restaurants, Happy Cow and Wild Cherry, if you're into vegetarian food. If you go there you might as well visit the Museum Of Childhood on Cambridge Heath Road by Bethnal Green tube station, it's not very big but it has lots of nice lod artefacts to look at and it's free. Other nice places in the Bethnal Green/Hackney area, just for hanging out, eating and drinking can be found on Broadway Market which is a very nice street by the canal. Oh well, once in this neighbourhood, you should go to Brick Lane, lots of cafes and nightlife and legendary curries if you like your indian/bangladesh-food. On a parallel road to Brick Lane (Commerical Street) you'll find the pub, Ten Bells which apparently was frequented by a certain Jack The Ripper, in fact this whole area is overflowing with Jack The Ripper and the Kray's references if you do your homework and look for them. Also, Whitechapel High Street is nearby, where you'll find the venue called the Rhythm Factory where bands like The Libertines played before (and after) they were famous and lots of other bands play regularly. You can also find the pub, formerly called the Grave Maurice, where Morrissey was photographed, but it's called something else now and isn't particularly interesting. Oh yeah, and a trip to the venue, Pleasure Unit, on Bethnal Green Road might be worth checking out as they often have good bands playing there, plus it's very close to Vallance Road where the Kray's apparently lived at some point. Bethnal Green Working Men's Club is also pretty good for a fun night out, either with or without live bands - I actually saw The Boyfriend's there in 2004 (not that i liked them very much).
Sorry for the long post but I'm quite enthusiastic about this area of London as I lived there for some years and am returning for the first time since last year this month
Other 'obvious' places to see include Camden Town - go to the Elephant's Head (pub) after the market has closed and most non-locals have gone home - it's just a pub, but it has a nice atmosphere and you can still catch a bit of a rockabilly/punk/skinhead scene there if you're lucky. A walk from Camden up to Chalk Farm/Primrose Hill is also nice when the weather's good and for a good view of London (maybe have a picnic there).
The Grapes pub on Narrow Street in Limehouse is lovely. And the Southbank near Waterloo is nice during the day if the weather is good - with the outside cafe at the NFT and the outside bookstalls and the Tate Modern nearby there's lots to do around that area as well. The last place I will suggest is the Notting Hill Arts Club, not surprisingly in Notting Hill, which I've had some fun times in (on Wednesdays and Saturdays, but I don't know what it's like now).
Get Time Out magazine and look at the listings for what's on. Hope you'll have a great trip