I’m going to admit now, because there are things that all of us editors have to look up (right? Please share! Better to check than be wrong) and this is one of the ones I have had trouble with in the past and have had to commit to memory. Which one means the most important and which one means the first performance?
Premier means the first in importance or order, and this extends to being used for a head of state (in Australia and Canada, it’s the official title for the chief minister of a state or government, everywhere else it can be used to denote the head of state, “The British premier stated that …”).
A premiere is the first showing of a film or the first performance of a musical work or play. So you go to a film premiere, not a film premier.
Premium has several meanings, one of which is superior and usually more expensive – premium cat food, for example (a premium is also the amount you pay for insurance or a sum added to a normal sum in order to get something better).