Enquiry emails for grad school
Most professors that I know in the US (in computer science) do not want to receive such emails and will ignore them. Many have a note on their website saying so. The expectation is that students should submit an application and only after they are admitted should they talk to faculty. If you are interested in a particular faculty member, you should note this in your application.
That said, norms vary greatly. In the UK, the expectation is the opposite in my experience -- you should only apply after contacting a potential supervisor.
You should look at the websites of faculty that you are interested in working with and see what they say about contacting them. If they do invite such inquiries, then you should contact them. In terms of timing, earlier is generally better, as long as you are at the stage of having reasonably concrete plans.