If you google your town's public school system and kindergarten enrollment, you will likely find a starting point there. As it gets closer to the spring, more information should emerge. You can also find out what the closest elementary school is to your address, and contact them directly and simply ask about the process.

As for head start, the doctor may mean preschool, but your town may have a specific head start program, which your town's public school department should have more information on.

It's okay to not know these things. I grew up in public schools, but this was all new to me as well when it came to my own kids. It's a learning curve for everyone, and you'll come to know all of these things, too :)