I know that Alberta gives you health coverage right away if you are coming from a different country but if you are moving from another province, they cover you starting the 1st day of the 3rd month since you moved there. Other provinces, you can find their individual health care systems websites at http://www.servicecanada.gc.ca/en/subjects/cards/health_card.shtml but in most cases you are covered after 3 months or less.
The BC health contact info is here: http://www.healthservices.gov.bc.ca/msp/infoben/contacts.html
If you want to get PR in Canada, you have to apply for PR. If you have enough points to apply as a skilled, you can do that. If not, you can look into doing a PNP program after you finish your studies.
