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.