To play the bonus rounds at any time, insert any game into your Sonic & Knuckles cartridge. When the "No Way" message appears, hold A + B + C and press Start. If you do this with the original Sonic game you can play the levels in order and will be given a password to save your progress.

Note: In case you're wondering, I've left out the passwords deliberately. There are over 32,000 bonus rounds in Sonic & Knuckles. I can't possibly list them all on the page, nor is there much point to doing so since the levels are nearly identical. Thank you.