I'm trying to get my bitcoins that I bought years ago, but my Bitcoin QT never synchronized to receive! I have now downloaded bitcoin core and I was informed that once I did it should automatically simply transfer over, but it did not! So what are my options to get them back?
After the purchase, you've most likely sent the coins to some address. If an address was from BitcoinQT, only thing it needs to do is synchronize, and coins will be in there.
However, you mention that "I have now downloaded bitcoin core" . If you use new BitcoinQT client, and especially on a new device, it has no idea of your previous keys, and it generates new ones.
To conclude, what you need are private keys that correspond to the address you've sent the coins to. Maybe try to import them in another wallet, like Electrum