Bitcoin Wallet is downloaded I think. What next?
Addresses are not private information. If you ever want to receive bitcoins you will have to give one of your addresses to the sender. 

The information that you keep private is what is known as the "private key". Each addresses has its own private key. The private key is what allows you to spend the coins sent to the corresponding address.

Usually you don't see the private key in your bitcoin-qt window. The private key is stored in a file called wallet.dat in your bitcoin-qt folder. If you decide to encrypt your wallet by setting a password it is the wallet.dat that is encrypted. So the best way to keep it safe is to a) set a password and b) routinely backup the wallet.dat file.