I'm currently in progress of changing my last name after marriage in Vancouver. These are the steps I followed changing my name in Vancouver, BC Canada.
BC Medical Services Card
This is probably the easiest to change. Call (604) 683-7151 and it only took me 5 minutes to update over the phone. Once you make this update, you'll have to do the next step.
BC Drivers License
Bring primary ID (something with photo) and a secondary ID and your marriage certificate. Request for the name change for your license and your medical card. If you phoned Health Insurance BC, the name change for the medical card happens faster.
Social Insurance Number
Bring your two new IDs from above, your birth certificate and your sin number. It will take about an hour at the Services Centre usually, so plan on that! You can find a location here!
I have not changed this yet because I have a couple trips booked in the next year. I've been told travelling in your maiden's name is OK as long as it matches the ticket and credit card that booked it. You can carry your marriage certificate to be safe. In order to change your passport, you have to apply for a new one.
Bank and Credit Cards
I also am leaving this til later because my own business. For the bank, you need to bring your marriage certificate, your new IDs and passport. It should not take long, but if you have investments or anything complicated, consider booking an hour appointment with a specialist there. It might be worth it to bring your partner if you have a joint account.
For the credit card, once your bank is updated, it is a phone call to whoever your provider is.
Car Insurance / Insurance
This would be most pressing after your primary IDs are changed and banking info, so update your provider ASAP. Bring all IDs, including your marriage certificate.
Everything Else (Social media, Amazon/Shopping Sites, etc)
I feel like I have to sit down one evening and make a whole list of websites I've ever used for anything and update it all, unless I don't use the service any longer. I'm sure my maiden name will still pop up from time to time, but the most pressing ones would probably be those that have billing associated with it especially after the banking info change.