Sonic & Knuckles

An easy one which can be achieved on the first level. Head to the first giant ring which should be fairly early on in the Mushroom Hill zone - as Sonic, Knuckles will run from a doorway that he then shuts - you can find it in there as both characters. When in the 3D special stage (similar to those from Sonic 3), you need to collect every ring by turning all of the outer blue spheres into red ones when faced with a squared ring of them. This will net you a continue as will collecting as many rings in the bonus stages as possible.

OR you can head to the bonus stages (the slot machine or the one with the glowing orbs) and collect as many rings as possible.

Use the save anywhere feature on the pause menu to make sure you don't waste your chance at getting a continue.

nice and easy this one, to extend on the previous solution in regards to the slot machine bonus stage, the required total is 50 rings for the continue.
it doesnt want you to get all 50 in the level, you can enter the bonus stage on 49 rings and collect 1 ring to gain the continue... also you can enter the stage with any number of rings over 50 and again, collect 1 ring and the continue is yours.

There are 2 ways in which you can earn a Continue in this game.

Option 1: Enter a Special Stage, turn the Blue Sphere's into Rings and grab 50+ of them. To turn the Blue Sphere's into Rings simply find a group of blue spheres which consist of at least 4 or more and then run around the outside. If done correctly all of the rings around that blue sphere will turn into rings. If you manage to get 50+ Rings in any Special Stage you will hear a chime sound, earning you a Continue.

Option 2: Grab at least 20+ Rings and then find a Checkpoint in any level. You will see some stars spinning around in circles at the top of the checkpoint. Jump to enter the bonus stage. When you enter the bonus stage (the bonus stage will be similar towards the Special Stages in Sonic The Hedgehog 1), you will see a Slot Machine (similar towards the Casino Night Stages from Sonic The Hedgehog 2). You will need to either use the Slot Machine or grab the rings around the bonus stage and after you get around 30+ Rings you will hear a chime sound earning you a Continue.

Regardless of which method you use, if you hear the chime sound you will get the achievement.