For Social Insurance Number (SIN) you need to find Service Canada office in your city/town and show up in person. You will need valid work permit and your passport. My SIN was expired so I brought my old SIN with me. You come up to reception they put you in the line and you wait. Then an agent will call your name, look at your work permit, passport, verify some info and that is it. You will get SIN in the mail after 10-14 business days.
Health card: I have never had one, so will just share what I know and will do tomorrow. I found an office downtown for Health Care Insurance Registration Service. I went there and a lady at the register gave me an application to fill in and asked my husband (who is Canadian and therefore will add me to his health care plan) to sign that application. What is more I will need to bring my passport and valid work permit. I filled in an application, my husband signed it so I will try my luck tomorrow, will let you know how it goes.
Hope that helps. And bear in my mind all provinces may have different requirement, I live in Calgary, Alberta.