It depends on your discipline and self-criticism. If you can truly see the fault in your own work, then yes that is a great way to start.

What goes hand-in-hand with this is going to look at photos that you find awesome (books, galleries, museums, even online gallery of famous photographers). This gives you a baseline to know what is possible and lets you work out why you prefer their photographs to yours.

Then, shoot a lot and criticize and compare your work to your own previous and that of those you admire. Repeat until you are no longer advancing. After that you will need something else to break your learning plateau: photography books, courses, seminars, etc. Depending on the style of photography you seek, photo tours are an excellent way to learn.

As for the camera. You can start with anything with full manual controls but it helps greatly if you get a more advanced model so that those controls are more efficient. You can always upgrade later but if you buy into a system right away, it gets harder to switch. On the compact side you can do a lot with a Canon Powershot S95 for about $350.

Honestly, most people who come to my photography classes having a bought a camera already quickly learn they do not have what is best one for them once they learn about photography and what each type of camera and lens is ideal for. So, I personally recommend to learn as much as possible before jumping into a big investment. Technology will improve in the meantime.