I bought 1 BTC from localbitcoins.com and received my bitcoin in my online wallet with no problems at all, after that I then sent it to my "bitcoin-qt wallet" address located inside the Bitcoin wallet itself under the "receive" tab. My bitcoin transaction seemed to work because the transaction information was on blockchain.info but once I looked at my "bitcoin-qt" wallet there where no bitcoins in my balance? Where have I gone wrong? Where have my bitcoins gone? it also says (out of sync) on the overview page and also "no block source available on the bottom left next to a bar saying "32 weeks behind"
If your client is "32 weeks behind" then it hasn't yet received the transaction that sent the coins to its address. Once it is in sync, you should see that transaction and a correct balance. If you are able to connect to the network, the client should sync automatically (though it may take many hours or even days, depending on the speed of your computer and internet connection). If it is not updating at all, check that your internet connection works and isn't blocked by a firewall.
1) You were correct in moving your Bitcoin from your online wallet to local storage - you want to control them yourself
2) You need your Bitcoin client (on your computer) to be in "sync" with the Network - right now it's synced up to 32 weeks ago. You want it synced to today. There is no button to sync - it happens automatically when the program is open. Note: I'd usually do this before sending any coins, to make sure your computer and network settings are correct.
3) For now, let your computer sync overnight or longer. This means leave your computer on, connected to the Internet, and your Bitcoin client open. Although it might not look like it's getting anywhere, it probably is.
4) If your Bitcoin client never gets anywhere (after two days), there is something wrong with your Internet connection or your computer. You can either move your computer (laptop) to another connection and try there, or ask a friend who you trust to look at your computer/Bitcoin wallet setup.
5) Do not trust people online (here or elsewhere) to troubleshoot your computer, as anyone with access to your computer may be able to extract the coins.
Remember: Your coins are now assigned to your computer's Bitcoin wallet - you need to get that wallet working or risk losing your coins. Next time, make sure the wallet is working before sending them there! Hope this helps.