I made some tests for you. If you start with the default type of wallet, which is a HD wallet, the backup means that it asks you to write down the seed, which is a list of English words.
You have to write them down in that order somewhere safe. The private keys of the addresses inside your wallet are all generated based on this seed, so you should not need any new backup.

1. The backup saves the seed of the HD wallet. Words to be written onto paper, make sure you don't mistake them. The backup never transfers Bitcoin.
2. No, you should not need new backups after every transfer.

Mycellium creates a HD wallet for you. When you played with it a little (even backup seed, write down public address, unistall Mycellium, restore from seed, verify public address would be interesting and good for the peace of your mind) then you can transfer your Bitcoin from the old wallet you have been using into Mycellium new address. Then you'll be ready to use it for real.