I also think it is important to identify professors who work with methods/populations/questions you'd like to research and talk to them. I think it's important to learn about the projects they're currently working on and about the work of other faculty members at the school. This will help you see you're a good fit for the program.
These talks also helped me a lot with my statement of purpose. Not only can you tailor it to the specific program you're applying to/professor you're citing in your essay, they also helped me figure out the best way to frame my research interests and how my CV relates to all that. So while it is good to start writing your statement soon in the process, I would also recommend keeping your mind open and not being afraid to change your statement (maybe substantially) during the application process.
One thing I would say about talking to professors, though, is that quality is more important than quantity.
I talked to a lot of people, and some of these talks didn't help me at all. One or two might have even decreased my chances of getting accepted, actually (lol). So spend some time looking up the professors you'll get in touch with. Don't contact people just for the sake of talking to someone at the school you're applying to. If you're not ready to have a meaningful conversation about how their research interests relates to yours and don't have any other good reason to contact these people, look them up first. My successful conversations were with people whose work I knew well or people whom had been recommended by professors I know well.