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Some great suggestions here.
Borough market is definitely a must see - the brownies there are world famous. Portobello market is kind of fun too (it's where Notting Hill was filmed) and Camden Market is a good visit too, it will be fully reopened after the recent fire by the time you come in August.
The Tate Modern is one of my favourite galleries, free to enter (optional donation) and I particularly like their Dali works. Lots of the museums and art places in London are free or very cheap which is a bonus.
I recommend looking up London walks especially during summer - there are scheduled times when a guide will take you on a walk which passes interesting historical areas. The guides are typically people who are fascinated with the history of the places themselves so you can get great insight on quirky landmarks. Doesn't cost much, and it's much better value and more fun than the tourist buses.
As for crime, most of the dangerous places are ones you wouldn't see many interesting spots anyway (Hackney, Brixton, Aldgate). I'd recommend general "staying safe" procedures: don't look like a tourist. Don't wear a "fanny pack", carry large amounts of tourist maps, cash, dress outlandishly, talk loudly in an American accent etc. American tourists tend to attract a lot of attention and in London it's generally best not to attract attention. Do those things and you will stay out of trouble.
Send me a PM just before you come over, I may have some time free to hang out and take you around. I always like exploring London when I can, it's so easy to take it for granted when you actually live here.
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