I'm not sure I understand all of the bitcoin storage options

I know there are bitcoin addresses and bitcoin wallets

what is the relationship of an address to a wallet? is a wallet mandatory and a given when talking about any bitcoin storage option?

if I had an account at a bitcoin exchange, how could I quickly move those coins from the exchange to the same wallet that I am using to dump the coins I mine?

I might not be asking the right questions, because I am not sure about the relationship between the various storage options of bitcoins (wallet, exchanges)

I would like to understand this