Most of the universities starting with "University of*" are likely to be better ranked than the others and are a little difficult to get in. However, you should not be afraid to contact professors and apply only when someone agrees to take you as their student or be your supervisor. That way you wont lose money by applying to places where you are not likely to succeed. You have to read about the research done by each professor before writing to them. The professors may not reply right away. You may have to follow up with them. Sometimes they do not reply at all, you should not take it personally. For each professor you have to write a separate letter highlighting the skills/interests that make you ideal for research in their field. All this takes time. If the program requires GRE, you will have to prepare for it for at least 2-3 months. Taking the English language proficiency tests also take time. Then there is the writing of statement of purpose (SOP) for applying. You have to write it carefully and get it reviewed by your friends, professors etc. You will have to contact your past professors for giving you recommendation letters. I think I have given you a lot of info to digest in a single post. Try reading about the research of the faculty first and post if you have any questions. Also click on "how to apply " link in those universities' websites.
