First off, you can photograph anyone anywhere in an area where they don't have what's known as a "reasonable expectation of privacy." This means any area accessible to the public, not including areas like dressing rooms, toilets, changing rooms etc. etc. Obviously private areas like homes, yards with hedges or fences tall enough to block site etc. are not "public."
However, publishing the photos is a bit different -- it's generally one big gray area. Using such an image for commercial purposes, e.g. to sell orange juice, requires permission from the model -- a model release. However, publishing it for editorial or artistic purposes is more or less OK, as long as the individual isn't portrayed in a bad light. Celebrities have less protection than other individuals, so while you would probably get away with selling snapshots of Britney Spears sunbathing topless (assuming she was on a public beach), you would get in trouble publishing pictures of your non-celebrity neighbor in the same circumstances.
The trouble is that there is no unambiguous rule about what's OK to publish and what's not. In general, if they're private individuals, the rules are on their side. However, street photography is a genre that's very much alive and well, and it couldn't happen if you had to get a model release for everybody every time. Most street photogs take their chances and publish or exhibit their work without asking the people portrayed (but keep in mind the bit about not showing them in a bad light). In practice, people very very VERY rarely complain; in fact often quite the contrary. (One very prolific street photog I talked to, who's shot maybe 50,000 frames over the past seven years and exhibited thousands of them on the Web, in publications, and in galleries said that he's been contacted twice -- and in both cases it was because the people in the picture liked the shot so much they wanted to have a copy.)
As long as you're not guilty of libel or slander (remember the bit about not portraying people in a bad light), you can't get into REALLY big trouble. I remember a case of a travel photog brazenly using a picture of just some (recognizable) people on a beach in a commercial brochure for a travel company. The people sued, and settled for half of what the photog got paid for the photo -- which ended up as about $150, IIRC: nowhere near to covering even their legal fees. So basically if someone does complain about a pic on your website and you agree to take it down, you can't get into too much trouble.
So, in a nutshell: yeah, you can photograph pretty much whatever you want, and yeah, you can publish or exhibit it on your website or elsewhere as long as (1) it's not for blatantly commercial purposes and (2) the people aren't portrayed in a bad light.
So, how to go about it? Know your gear so you can use it confidently and easily, be casual, friendly, and open about it, don't be shy, participate and interact with the people you're photographing, if someone catches your eye, smile; if someone asks you what you're doing, explain. It might not hurt to keep some samples of your work around if that'll help you explain yourself, and it won't hurt if you'll offer to delete the photos if they're bothered by being in them.
Petteri
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Me on photography: [ http://www.prime-junta.tk ]
Me on politics: [ http://p-on-p.blogspot.com/ ] 
