My high school is well known for it's ability to assist students through the application process, so I can help you out on this as they basically drilled everything into our heads last year. This is what we did:
You'll need to address and stamp the envelopes (you should use those big manila envelopes so the contents are nice and neat) before you give them to your guidance department. The address would be the university's office of admissions, which you should be able to find online. The return address should be your high school's address (with your homeroom number).
The only application I did online was Rutgers, so the envelope that I gave the guidance department just had a transcript request paper in it to let them know all they needed to do was mail my transcript in the envelope. I printed the transcript request off the Rutgers website.
All my other apps were paper apps. You should be able to download them from the university's website. For example, one of the schools I applied to via a paper app was the University of Michigan.
This is what the paper app looked like for them. Just print it out (use a laser printer if you can) and fill it out. Then, put it in the envelope and give it to guidance. They'll put your transcript in and then mail it out.
Also, some of the school's you might apply to may use the
common app. I
printed out my common app's, but I think you can do them online. I never bothered to check that out though. I preferred the paper apps.
This is a list off all the universities that use the common app. You'll need to print out one copy for each school that you are applying to. So, for example, I applied to Cornell, Northwestern, and Carnegie Mellon, so I printed out 3 common apps.
Your counselor should go over this with you in your meeting.
And fwiw, I went to SUNY's website and I think you can apply online if you want.
Do you take care of your teacher recommendations yet?