Rings can be used as portals or power stuff. Chaos emeralds are useful most of the time. It can use as a power source, chaos control, goes super transform, etc.
Ring
Ring can be used for portal or power stuff (like Sonic X) and it did bring Sonic metal (not Metal Sonic) version back to normal Sonic.
Chaos Emerald
Chaos Emerald can use for power (like chaos control for example). Note: if you got all 7 Chaos Emeralds, then you can go Super State (like Super Saiyan).