Lots of chain slack... Time for a new chain? Bad derailluer? I got home from a short ride up the street today where there was a lot of clicking and skipping of chain links and when I got home I realized my chain had a ton of slack. After looking around on the internet, I tried moving the rear wheel back, but there's more slack in the chain than there is room to move the rear wheel back, so I'm stuck. 

A cursory understanding of bikes and reading the internet suggests that it could be a stretched out chain or an issue with the derailleur. This is an old steel 12-speed Raleigh. Granted, I don't take the best care of it (not sure where I'm supposed to wash it and store it - I live in a small apartment), but I do depend on it.

Here's the picture of the state of things:
It appears your rear derailleur pulley cage spring is not doing its job. In your picture, it should be pivoted back towards the right side of the picture, maintaining chain tension.

You'll have to remove and clean out your pulley cage, clean up and lubricate the spring and reassemble it. If that doesn't solve your problem, you'll have to replace the derailleur. You can see how to service many brands here: http://www.parktool.com/blog/repair-help/rear-derailleur-overhaul