I have about $200 in bitcoin but my wallet (Core) won't synchronize the blockchain (it's been doing this for days) since for some reason it got corrupted (As it tends to do say every other month) and now I'm unable to retrieve my money. I want to transfer it to a lightweight wallet so this won't happen anymore but I don't want to lose the money I have locked in there. How do I transfer this money when I can't synchronize to the current state of the blockchain?
You'll want to use bitcoin-cli to dump the priv key and import it into a lightweight wallet. Here's the command to do this in bitcoin core:
$ bitcoin-cli dumpprivkey <address that contains your coins>
This should dump out the WIF format private key.
You can import this private key into something like Mycelium which will then have access to your funds. Typically, the wallet will sweep the funds from the address containing your coins to an address controlled by Mycelium. In any case, you'll have full access to your funds after you do this, though at the cost of some miner fees (about 10 cents)