In Sonic the Hedgehog 1, there are 6 Chaos Emeralds you need to collect to get the good ending, you'll need to collect 50 rings or more through the stages and finish the stage by entering a Giant Ring that will lead you to a special stage you'll have to explore and get the Chaos Emeralds. Save States are allowed for the achievements, so if you're having trouble with them, just load and save until you get them all.