Paths:
There's usually at least a high path and a low path. High paths are usually less treacherous with extra rewards but relatively hard to get to and relatively easy to fall from. There's stuff hidden all over the place, so there's no hard and fast rule about which way is best. Unlike Mario games, Sonic games aren't really designed to give you the opportunity to explore everything, so it's all about trying to get the ideal performance out of each run.
Rings:
Others have said this already, but you need at least one to take a hit from an enemy or hazard, 50 to access a special stage (needed for 100% completion), and 100 to get an extra life. So it's beneficial to get as many as you can, but sometimes risky or difficult to do so, just like in Mario.