The Bitcoin-Qt wallet from bitcoin.org isn't usually the best choice for a newbie. You might want to take a look at the first two-and-a-half minutes of
this video to learn how to set up a web hosted wallet at
https://blockchain.info/wallethttp://www.youtube.com/watch?v=zSmxbtFQlmgYou also might want to look into Electrum or Multibit if you prefer not to use a web hosted wallet.
If you choose to stay with the reference client (Bitcoin-Qt), assuming you have enough hard drive space it should eventually finish synchronizing. Depending on your computer it could take as little as a few hours or as much as a few weeks.
You should already have a wallet address under the "Receive Coins" tab. You can create more addresses there if you like. Any coins that are sent to those addresses won't show up in the wallet until it finishes synchronizing.
Yes, if you save wallet.dat, you can restore the wallet. The wallet.dat gets updated with new addresses every time you create a new address in the "Receive coins" tab.
It also creates a new hidden address that it doesn't show you every time you send bitcoins anywhere. You need to make sure that you have
a recent backup of wallet.dat to restore. If the backup is too old, it won't have the new addresses.
