Remember, even birds that come out of the cage to socialize spend a large part of the day in their cage. Bigger is better as long as the spacing between the bars is narrow enough to prevent injury if the bird tries to escape. The bird's head should not be able to fit between the bars. The door needs to be large enough to comfortably put your hand through, catch the bird, remove the bird, and replace the bird. The latch on the door needs to be escape-proof also as the bird is going to have a lot of time to find a way to open it.
Some owners place a clip or a padlock on the door of their escape artist's cage.
Budgies like to move around and should have a cage that is tall and wide. Canaries and finches like to fly and should have a cage that is wide and long to allow for flight. Cockatiels need a cage that is big enough so the crest on the head and the long tail fit without being crushed. The size recommended at most pet stores is going to be the minimum size for that species of bird. Your bird will be happier with a larger, more spacious cage.