Clean your chain with some mineral spirits on a rag. Wipe it dry. Put the tiniest drop of Triflow that you can manage on each link. Let it set overnight and try to wipe it all off. Don't worry about the rust. If it's just supreficial, this process will make it go away.

Do do your shift cables, shift into your biggest sprocket. Now WITHOUT TURNING THE PEDALS work your shifter as if you were shifting into the smallest sprocket. That'll give you enough slack to free your cable housing from the stops on the frame. Clean them with some mineral spirits on a rag. You can lube them with some triflow if you want, but don't overdo it. I'm not convinced that it's any better than just clean dry cables.