It used to be a basic rule of photography: you cannot make a photograph without a camera. So bring your camera along, get used to having your camera with you, overcome any shyness you have about having a camera on you.
These days more photos are taken with cell phone cameras than with traditional cameras, I’m sure. So having a large DSLR around your neck seems to make you stand out in the crowd–for good or bad. And while cell phone photographs are essentially opportunistic, and despite some luck and/or great skill, tend toward the crappy photo end of the image spectrum, they don’t seem to fit with a more serious attitude to making photographs.
But back to the rule. IF you are serious about photographic quality as well as expression, and you prefer to use the big tools (DSLR and interchangeable lenses) then developing a habit of taking your camera with you will increase your awareness and help you see more photographs. I try to take my camera with me to work every day. Just the body and one lens. Even if I don’t use it, I walk about looking at my surroundings more “photographically.”
As for what type of camera you need, the answer is always whatever you want and can afford. Many wonderful images have been made with cell phone cameras, pin hole cameras, film cameras, compact cameras, and professional grade DSLRs. It IS all about the vision and skill of the photographer and NOT the equipment, but most people serious about photography end up wanting a DSLR with interchangeable lenses because they offer greater creative control over the final image. Short of being a professional photographer, most people have more than enough equipment of a very high quality, and could do just as well with less equipment. I certainly have way too much equipment. Given that few people see any of my photographs, and I print very few of them, I don’t need a Canon 1D X. But it’s certainly cool to have one! The advice I give most people is to let the use of the resulting images determine what camera to buy. You may find yourself wanting more, and better, but if you print 5x7s and show them once to friends before putting them in the shoe box for eternity, or if you post photos to your Facebook account every now and then, having a DSLR is probably overkill. If you want to print high quality 16x20s and sell them, you need a more expensive DSLR with good lenses. Again, most photographs are made with cell phones, and for good reason–they are available, easy to use, quick to use, and the resulting image looks OK on a 3″ diagonal screen…