As far as the types of pictures you should take? Anything. It doesn't have to be artistic or taken with any grand purpose. But you should use photography as a way to simply document even the most slightly unique things you see in your world whether they're visual curiosities, a funny moment, you eating somewhere new, the arrangement of clouds in the sky... anything. And what you'll find when you start sorting through your photographs, you actually will have an opportunity to recount all of these minor fleeting moments which you may would otherwise have forgotten.
We sort through our photos using iPhoto. We make albums based upon the content of the image such as "visual curiosities" "typography" "food" etc... Once we have sorted the photos we go through them, sometimes editing the ones we like, and then uploading them to our Flickr account for us to share with family and friends. It has become a great social tool as well, spreading daily journeys and interests with your friends.
Afraid of going at it alone or do you think you could use a little support? Get your friends together and work at it together. Create a Flickr Group for you to all gather your photos together. Alternatively, we recently started a photo club at our work with weekly assignments for things that we need to shoot and then share on Flickr. This was a big motivational tool in getting us out there and taking pictures.
What kind of camera do you need? Well... anything. It really doesn't matter. A great resource we blogged about recently is the short book A Lesser Photographer. It speaks about how unnecessary a "good camera" is to being a good photographer. Don't be afraid to use a point and shoot, your camera phone, or even your laptop's webcam! Giving yourself these kinds of limitations will often force you to think more creatively and innovate.
A popular photography project that many have been doing is taking one photo a day for a whole year. There are countless apps and even a website dedicated to helping people achieve this task. Although there is nothing wrong with this, our approach is a bit different. We don't feel an obligation to take photos every day. Some days we just don't see anything worthwhile and we don't think you should pressure yourself into bringing your camera places when you don't want to. So think of this less like a "project" and more like a lifestyle change. It shouldn't be uncomfortable or feel like a chore. It's about helping you appreciate the world around you by focusing a more critical eye on what you see.