Can I get promotion points as an E-2 and if so should I talk to my battalion S1 about how to see my points?

I really appreciate the advice this will help motivate me to get to E-5

You can begin working on things that will award you points. APFT performance, weapons qual scores, awards, military schools/courses, civilian education, etc. Ask your NCOs for guidance on how to complete things like these. If you have not maxed PT, start working on it. Your command can authorize TA so that you can do college courses. Perhaps your unit has some slots for Air Assault or Airborne that they can send qualified soldiers to complete. All of these things will help you later on down the road with promotion points.

You can start racking up the points but you won't be able to see them yet. Work on your college and correspondence courses so that when you're eligible to go to the board you aren't waiting around on points