How do I save a contact to a specific iCloud account? I have multiple account sources for my Contacts (family iCloud, personal iCloud, Exchange, etc.) When I create a new contact, how do I specify what account I want it saved to? Right now, Contacts just saves it to the default account.

I'm on the latest Yosemite.
Short answer, you cannot change it on the fly on a per entry basis.  You have to set the default account in the preferences for contacts.app. However, when in Contacts, and you have a card open (perhaps the one you JUST made), pressing the OPTION key will highlight all the groups on the left side of the app that include this card.  When you have your list of cards showing on the right with the accounts or groups showing on the left, you can simply drag that entry to the group or account of your choosing. This is the only method I'm aware of.