My friend's paycheck is a different amount than mine and he is an RA, too. Aren't the amounts supposed to be the same?

No, the paycheck amounts can be different. The amount depends on withholding, benefits, start date, end date, and semester salary amount that your PI is paying you. Therefore, it is rare to have the exact same net pay as another person.

Where is my first paycheck?

We most likely have not received a student appointment request from your professor. Contact your professor.

Here is the website where you can go to check the status of your paycheck:
https://www.asa.stonybrook.edu/hr/rftrackingsystem

Login to Docusign. Has your appointment been completed – signed by everyone including HR? If it is pending HR’s approval do you need to go there and show your ID?