Hi S.,
As a couple other mom's have shared, you want to contact the ECFE (Early Childhood Family Center) in your city right away and get an appointment set up (some are booked 6+ weeks out). Again, this is not a pass/fail test, as noted, it is merely a way to determine if a child could benefit from additional services provided by the state (i.e. speech therapy, etc.).
They test fine and large motor skills, eye and hearing test, memory, recognition, etc...
The screening is from 3-1/2 to 5 years old, but they really try to stress to parents getting in between 3-1/2 to 4 so that if your child were to benefit from some additional services, you could be using them now rather than waiting until kindergarten.
Definitely contact the school(s) of choice and find out about registration. Many have open houses scheduled in Jan/Feb, though you can also make a separate tour with the principal if you cannot make the open house.
The school can also refer you to an early screening center if you cannot find one.
Once you register to a school, some may have another screening just before school starts to determine what your child knows and what he may need help with at the start of school.
My daughter had the ECFE screening and then the late spring before Kindergarten the school did a short assessment to see if she needed to have help with basic learning (i.e. ABC, 123, colors, etc...). The school was offering a "no child left behind" program to get kids that perhaps had not had any formal preschool caught up with the other kindergarten students, so they were offered full-day kindergarten at no charge.
This can be a very exciting time, but I had my own trepidation about sending my first to kindergarten -- I didn't know the teachers, parents (many of the preschoolers we knew were going to other elementary schools in the city).
Try to get involved -- volunteer to help out once a month in the class. Attend the PTA meetings as you will get to know parents of older students so you can ask questions about upcoming teachers and learn about learning opportunities, etc.... Even if you work full-time, try to get you employer to let you take 90 minutes off work to be in the class (if done once a month, it would really be a minimal amount of time and you might be able to schedule it as part of your lunch). Your son will love it, you will get to know the other kids and it is at a time when your son wants you around.
Before you know it, he will be in middle school, not wanting to have much to do with you...
Good luck