Basically, speedrun them. Train on a save in an easier difficulty setting (preferrably Normal) and aim for the diamond medal times. Watch tons of Sonic game speedrunning videos to learn basic strats such as loop cutting or diving into lower sections.

Don't be afraid to die and blunder on purpose in your training save. You have unlimited lives in story mode. Sometimes doing crazy things might let you find completely unexpected shortcuts and strats. Know your territory, especially the bounds borderlines of the stages.

Ignore as much enemies as possible. Ocasionally, you might need to parry to cut through.

If you are doing step 1 correctly, it is expected your only hard time will be at the bosses.

You only have two hits before you die in Challenge Difficulty. Although it is tempting to rush boss fights, don't worry if you need to take longer. Train in an easier difficulty save to figure out boss patterns and ideal parry timings. As mentioned in the achievement guide, "if you're unsure, run away from the boss" and then attack whenever there's an open opportunity.

Even though the parry lasts only an instant in Challenge Difficulty, it is fundamental to learn an advanced strat, which is parrying during a homing attack (essential if you have to suddenly fight weak-type enemies such as spearbots, spikebots and knightbots). Timing a perfect parry when homing into a spikebot also destroys nearby weak-type enemies.

For maximum mobility in stages, use both Edgy Jester and Power Armor.

Edgy Jester lets you run with reduced friction and jump great heights depending on your current speed, along with three air dashes (great for recoveries). Well timed charges can even let you fly higher after climbing walls! Frame-perfect dashes during homing attacks may let you fly really long before you bump into something.

Power Armor has a longer dash along with a momentum-conserving area-of-effect heavy attack. It is also great for boss fights as it has a balance between mobility and attack damage, dishing out even more than Electric Jester.

About to die in a boss fight? If your power bar is filled enough (through parries and attacking) to display "3x damage", you can make a charge by holding RB on a XB1 controller or R1 on a Dualshock 4 controller. Once you have a big glowing charge on Fark, release to make a special attack and heal yourself in the process.

Challenge Difficulty works similarly to Sonic's ring system. If you get an H-Capsule, some Bits or heal yourself through the special attack, it's like if you have rings in possession. After getting hit, you die if you get hit again with no health/rings.

Play the second story of the original Spark The Electric Jester, especially on Hard Mode. There, the parry works just like in Challenge Difficulty, lasting an instant and no holding allowed. Later stages tend to have enemy floods to get an idea of difficulty regarding parry timings, a good spot for some practice.