After a long hiatus from any form of workout, I'm attempting to scale myself back into a regular bodyweight routine. The only issue I'm having is with Dips. I don't have access to a set of parallel bars, but I do have access to rings. The issue is I'm not quite strong enough to manage dips on the rings yet, so I'm not sure where to begin in order to transition to rings dips. What do you recommend?
You want to start with the support position, or what would be the top of the dip. You can move your hands away from your sides a bit, which will dramatically increase the leverage on your shoulders. As an aside, if you go far enough (with years of training) and you can even hit the "iron cross".
I would practice the support position, being able to hold that for maybe ~10 seconds at a pop. Once you can do that, start dropping a little bit, letting your elbows go up/back/out. It will be pretty natural once you have the support position dialed in and have moved up and down a bit to do a real dip.
The hardest thing about ring dips (as you're probably experiencing) is the shoulder stabilization, but once you get it I think you'll find them much more comfortable than regular dips. Mark Rippetoe has stated (Starting Strength v3, I think) that weighted ring dips are pretty dangerous, and I'd agree. The loads on your shoulders can be immense if you flare your hands out at all. So there's still a good place for regular dips if you want to pile on weights.
But good work on getting some rings, there are some awesome things you can do with them. Check out Beast Skills for stuff like the front plank, Jim does a great job of explaining those. Between front/back planks, dips, l-sits, and pullups they really are a fantastic investment.