I’ll try @Epanoui! Hopefully some of this is useful! A few to get started (note: I revolve around food... 
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A lot of people head to Brick Lane for Indian food (most restaurants are actually owned by Bangladeshi families). There are some really nice places, but actually it’s a bit of a tourist trap/attracts the crowds looking for a curry and a pint. Kind of like Chinatown - some fantastic restaurants there too, but hidden among overpriced ‘crowd favourites’ aimed at tourists. Anyway, head to Whitechapel instead. There are some really great Pakistani and other South Asian restaurants that really are delicious. Places like Lahore and Tayyabs. However they can get unbelievably busy, I haven’t been in a few years as a result but friends still love popping over there.
Columbia Road flower market is an old favourite, grew up going here on Sundays, lots of nice flowers to look at (and buy of course, if you fancy!) but also nice cafes and interesting shops along the side. Jones Dairy is popular.
I’ve seen the east end change a lot over the years so some of my old favourites are now hipster hangouts but some are still really good nonetheless. Brick Lane Beigel Bake is the best place for a salt beef beigel, and you get classic service - know what you want before you go in and queue, they work fast and don’t take kindly to dawdling once they get to you! They sell lots of other lovely baked goods, like onion platzels and challah bread.
On Broadway Market there’s still an original F Cooke pie and mash shop, still owned by Cooke’s family and run traditionally. I’m not sure if there are others but that one is definitely an original, they still make all their own pies, mash and liquor and jelly their own eels.
On Bethnal Green Road, Pellicci’s is a typical east end cafe that goes way back, I think the same family has owned it over 100 years. Perfect for a fry up, but they also do ciabatta sandwiches, homemade pasta and, I personally think amazingly for an east end cafe, affogato!
I feel like I should do some non food stuff
The back streets/parallel streets off of Shoreditch High Street and Hackney Road make for interesting strolling - that includes Columbia Road, but also places like Arnold Circus, plus these days places like the more popular Redchurch Street. You can sort of walk from Columbia, down to Arnold Circus, and then down to Redchurch through these parallel streets (parallel to the high street), and there’s lots of boutiques, restaurants, cafes etc. Interesting if you’ve not been before. Some really good places, these days it tends to be on the pricier side but there are still reasonable places to get a nice meal (and there’s a lot to choose from). Just realised I’ve gone back to food!
There’s a Hawksmoor church in Spitalfields if you’re a fan of architecture. On a nice day, a walk along Regents Canal can be really interesting. It runs from Limehouse in the east to the north, through places like Kentish Town. You walk past sites like old gas works (I find these structures quite fascinating), but they also tend to run through points with cafes etc. Broadway Market for example is also by the canal. There are also narrow boat tours you can go on.
A poster up thread mentioned Geoffrey Museum, which is a sort of interior design museum housed in former almshouses. Has a lovely herb garden (although as a pp mentioned, it’s under renovation until 2020). Bethnal Green also has the Childhood Museum which has been there for ages, it was acquired by the V&A some years back so it’s had quite a bit of investment in it too. Shows you all manner of toys and games over many generations, plus they do a lot of children’s activities too (but I still think it’s a fun nosy without kids!).
There are lots of walking guides you can download online. I think Limehouse Basin to Stratford is quite interesting (there’s a guide here).
I would say, also, Shoreditch is super popular. It has changed dramatically since I was a child. A lot of people love to visit, so I should probably recommend it! I don’t tend to linger on it as I’m not a fan tbh. It’s incredibly crowded, there’s a lot of drinking, smoking, drug use of a weekend (in the evenings, I mean!), however, if you’ve not been around Hoxton, Brick Lane, Spitalfields market then it is worth checking them out and seeing for yourself as there is loads going on and you can sometimes find great things in the markets. Sunday is a very popular day as everything is happening then. Some people love the buzz. But I grew up here when it was much quieter and less developed so I find it a bit overwhelming! (I sound like such a nimby) but lots of people love it, and to be fair when I was growing up it was very run down and a bit scary
Maybe I should stop there? There’s loads more I’m sure, like small art galleries in and around Hoxton, and also think walking along the River Lea around Hackney Wick is a nice walk. The Olympic Park also has some remarkably nice and quiet spots, away from the stadium side and towards the Velodrome, they’ve done a lot to create wetlands for wildlife and wild meadows, it’s lovely on a sunny day and again overlooking the River Lea.
And Victoria Park is a favourite, I suppose it’s just a massive park! But it’s very pleasant, they also have a lot going on (tennis courts, cricket areas, etc), good kids play areas, and lots of ponds and some nice cafes. Every once in a while it’s dominated either by a run or some sort of weekender concert but the majority of the time it’s remarkably peaceful and very green!
Ok, now I’ll stop...