I've been looking at the familiar feat, and I've been wondering: Can a familiar talk? If not, would the spell Awaken(give them human level intellect) make it so they could?

no, and no

Dragons are different. They have their own language but often learn to speak common.

Kay then. I know the familiar would be able to 'speak' with the master to an extent. Wasn't sure if they could converse with others.

Awaken & telepathy might work. Give it humanish inteligence, and over time it could learn common. It wouldn't be able to speak it unless it had vocal cords suitable to human speech, but telepathy would get around that. Or Telepathic Bond. It's more expensive, but would allow you to silently communicate with the familliar if it did learn common.