Asiana Airlines flies between: Vancouver - Toronto, Calgary - Vancouver, Ottawa - Vancouver, Vancouver - Calgary, Vancouver - Ottawa, Edinburgh - London, Vancouver - Newark, Newark - Vancouver, Sydney - Christchurch, Christchurch - Sydney, London - Manchester, Osaka - Sapporo, Sapporo - Osaka, as well as several other routes.
Thankfully, if flying to or from Canada on Asiana, you will be able to check two bags for free. They can weigh up to 50 pounds per bag in Economy class and up to 70 pounds per bag in Business class. If flying to destinations outside of the Americas, your allowance for check baggage may be different. If you can't quite fit everything into those two suitcases, you will be allowed to carry one smaller bag on board that weighs up to 22 pounds as well as a small personal item such as a camera or laptop.