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?

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

SGT (Join to see) Google Promotion Point Worksheet, first link is to the HRC app. Unfortunately the RP graybeards are not allowed to cross the CAC seal.

I can not vouch for this link, but it appears to have the criteria. I can't see the S1 doing a promotion point worksheet for an E2 that has two more promotions based on TIG/TIS http://www.ncosupport.com/armypromotions/promotion-point-worksheet.html

The awards/decorations earn you points
Military and Civilian education Earn you points

The Weapons and APFT score are perishable and will be loaded off most current when you do the worksheet