I got pinned yesterday, 1 September 18, and I'm trying to get SSD2 done soon as I can. Do I have to get with S1 to get enrolled?

I don't really know the answer to your question; when I was doing SSDs it was something that you could just enroll in over AKO and get it done - I had SSD1-3 finished in no time.
I just wanted to say congrats on getting your chevrons and good work jumping into the SSDs without being prompted to do so. It is always better to knock those little things out long before they ever become an issue.

No. What the typical thing that happens is that you should be automatically enrolled once the Army system as a whole is tracking that you got promoted. Again, that is what is usually supposed to happen. But, if you want to get ahead of the game, then you get with your Training NCO and have them submit an application.