Street Photography was born with a very long, yet regular exposure for the time picture at Paris. This image is often referred as Boulevard du Temple and it was taken by the hands of Louis Daguerre in which a person stood "still" for the amount of time the exposure took. Daguerre wasn't trying to capture the person, it was just a serendipitous found object, and after that moment in time, street photography was born.
Street Photography is one of those genres that have been polemical due to people and photographer's judgment deciding whether something fits its scope or not. In other words, you'll encounter a lot of "street photography police" out there judging and arguing if your images are street photos or not.
Personally I have defined myself as a Social Photographer, and my statement clearly speaks about it. I also believe that "street" photography shouldn't be limited just to the streets, since it serves as a door for crafting deeper and meaningful projects around certain subjects, but that is called many times Documentary Photography, and we'll leave it for another occasion. Street Photography is not exclusive of streets (which are great findings of urbex) but also can be done in rural contexts.
Without further ado, let's talk about the basics around Street Photography.
Street Photography glorifies with the appliance of candid approach to people, especially strangers. Candid is not a style of photography, it has a larger span because it is a way of doing things. In summarized words, shooting candid images is about getting invisibly involved with societies and crowds with the primordial purpose of capturing meaningful stories that otherwise will remain unseen to the public. Let's also make a separation between diving into a crowd and start shooting like crazy. Shooting candidly requires attention and must not be left to randomness.
Keeping a Camera always with you
I love giving advices to people every time I have the chance to, and I love receiving advices as well. The best one I could offer you, is to always have a camera with you. The feeling of remorse and regret that I have had thanks to not having a camera with me in moments that I can only remember (because I have no pictures) is something I just wouldn't wish to anybody. There is a peculiar image that lives in my memory, wandering and haunting me with its vivid image that got exposed in my mind, but I can't show. Get yourselves a trusty camera that can be with you at every time, and you'll enjoy Street Photography way more.
Many times I've got commentaries stating that "why bothering having a camera always with you if you can take impressive images with a cellphone", and it is true. But personally I think that having a camera with you bugs you into being aware of your contexts and surroundings in order to avoid moments slip out your sight. In the other hand, by having a phone, your guard could be less awoke because having a cellphone with us is so natural now that it doesn't reminds you to be aware of the things that are happening around you.
Not everything should be Black and White
We have had the wrong idea that Street Photography must be done in black and white formats. Who said that? Who stated that as a standard? I love monochrome conversions in digital and shooting in black and white when I have the chance of shooting with film, but that doesn't mean street imagery should be done only with black and white. Only tweaked sensors and this pricey friend are capable of shooting monochrome natively in digital, all the other stuff comes by post-processing. You should decide to go black and white only if your color image has the potential of looking better in monochrome format.
I love Black and White, and I know for sure that it removes the distractions that color depict, but there are tons of great street photographs taken with color. A couple of names like Saul Leiter, Steve McCurry and Jeff Wall comes to my mind. All of them are known for their specific niches in photography, but they have taken images that cope pretty well with the genre and definition of "street photography".
Ok, this is something you really don't "need" to acquire, but they will make your life a lot much easier as a street photographer. If you are willing to get yourself more committed to Street Photography, it will be very likely that you'll be capturing images of people in the majority of cases. People feel threatened (and it is obvious) when you point a bazooka lens attached to a huge DSLR camera in front of their faces. Just imagine walking around your local city, minding your business, or just looking for something to eat, when suddenly a street photographer is aims at you a 70-200mm lens to your face. It will be very uncomfortable. Now, imagine the same scenario, but the photographer was so stealthy, that you didn't even noticed him so much, and you even thought that he was just some regular amateur shooting some snapshots. Now that character is the one we need to perform in the streets. Forget about that "I'm a pro" look when doing street photography. Another thing that is important; right the first second a photographer stumbles into a crowd with a massive camera and lens, it will get easily spotted by people, and they'll start to feel a little bit edgy, especially if such photographer starts shooting picture right the first moment he got close to that crowd.
Don't worry about the "Oh you are not a pro, you're shooting with a tiny camera" comments from people. You better start worrying about the images and moments you capture instead of the Pro looks.
I'll tell you a little bit about my digital gear background. I entered photography through a small P&S camera. I got mugged while doing some street photos and I lost it. I loved that camera, but since I was getting myself serious around photography, I got myself an entry level DSLR. I bought various lenses, and my favorite lenses for the streets are (or were, since I don't use them so much for street) a 28mm, and a couple of pancakes (a 40mm and a 24mm). After some years working with the DSLR, I went back to small gear; and got myself another Point and Shoot for doing my beloved street photography. I just wanted something that I could carry on with me always. After two years of that, I got myself the first "Digital Rangefinder" camera in history, and I don't plan to go back to those chunky and noisy DSLR cameras for inconspicuous photography, it is just nuts to try, and trust me, I did a lot of street photography with my DSLR, now I use it only for commercial purposes.
Mind your Social Skills
Being able to blend within the crowds, and small packs of people is crucial in order to achieve meaningful street photographs. This will vary from person to person, and I can only say that whatever you do on the streets and society, always be respectful. It doesn't matter if you don't share the same moral, religious or even political values from one place or another, please be respectful, and if you think you can't handle it, please use your camera for another niche of photography. My intention is not to be rude, but this is the thing I love the most doing, and I need to do my thing in order to keep it clean from people that could mess it for being rude with other people at the streets.