There are a few things that you might want to know before looking into Challenge Mode, and that is mainly the ranking system. The ranking system isn't all that useful to know about unless you're going for the leaderboards, but it is important in a few select areas.

Secret Exits

Secret Exits are a requirement to unlock secret paths, usually for a warp to somewhere further up in the map, or for an unlock. To get the Secret Exit, you must complete the challenge with at least an A+ rank, which varies from challenge to challenge. If the challenge is just getting to the end of a level with an effect such as Disco Lighting/Retro Fuel disabled or something, it usually means that you have to get 100% accuracy, with or without Retro Fuel. It's pretty impossible to go into specifics as it isn't really explained in the game, so just try your best at a challenge and you should get it to unlock!

Warp Locations

For those wanting to get around the map quicker! Be warned that you can't backtrack to earlier areas if you wanna go back to other areas though.

Secret Exit of A1 > F1
Secret Exit of C2 > H4
Secret Exit of F2 > Meat Boy Player Ship + K2
Secret Exit of H5 > N1
Secret Exit of K2 > P4
Secret Exit of N2 > S2
Secret Exit of P5 > U5
Secret Exit of X2 > Y1, X4 and Y3

Regarding Specific Unlocks

While I can say for sure where the Cheats and Player Skins are located, the order you unlock the Image Sets in doesn't really matter as collecting one just unlocks another set in the collection. As for music, not sure if it unlocks the same way as image sets but it doesn't specifically say what songs unlock on the World Map so I can't really be sure.

With all this said, let's get on to the actual guide!

Big Head Mode is unlocked by clearing E5.
Lane Mirror is unlocked by unlocking the secret exit of P3.
Disable Row Coloring is unlocked by clearing P1, which requires the secret exit from N4.
Disco Mode is unlocked by clearing J3, I3 or J4.
Flip Screen is unlocked by clearing either Y2, X5 or Y4, all of which require the secret exit from X2.
Retro/Noir is unlocked by clearing T3, which requires the secret exit from S3.
Stealth Mode is unlocked by clearing Z2 or Z3, both of which require the secret exit from X2.
Turbo Mode is unlocked by clearing Y4, which requires the secret exit from X2.
Zoom Mode is unlocked by clearing W4, which requires the secret exit from V2.

There is a total of 9 artwork sets in the game, and this is a list of all of them in order of appearance:
Clearing B1, which requires you to unlock the secret exit in A2.
Clearing E2, which requires you to unlock the secret exit in C3.
Clearing E3, which requires you to unlock the secret exit in D1.
Clearing G5, which requires you to unlock the secret exit in F4.
Clearing J3, which requires you to unlock the secret exit in H4.
Clearing M3, which requires you to unlock the secret exit in K4.
Clearing R5, which requires you to unlock the secret exit in Q3.
Clearing X5, which requires you to unlock the secret exit in X2.
And clearing Z5, which requires you to unlock the secret exit in X2!

Like the artwork, I am pretty sure that the order you unlock these in doesn't matter. So here's the list in order of appearance:
Clearing B4, which requires you to unlock the secret exit of A3.
Clearing D3, which requires you to unlock the secret exit of C1.
Clearing C5, which requires no secret exit. Huh?
Clearing G4, which requires you to unlock the secret exit of F4.
Clearing J4, which requires you to unlock the secret exit of I2.
Clearing M5.
Clearing O4, which requires you to unlock the secret exit of N2.
Clearing P4.
Clearing R4, which requires you to unlock the secret exit of Q2.
Clearing T5, which requires you to unlock the secret exit of S4.
Clearing U1, which requires you to unlock the secret exit of S4.
Clearing W3, which requires you to unlock the secret exit of V1.
Clearing Y2, which requires you to unlock the secret exit of X2.
Clearing Z2, which requires you to unlock the secret exit of X2.

And now for the Player Ships!

Dread Rick

Dread Rick can be unlocked by clearing S1.

Armored

Armored can be unlocked by clearing L4, which requires you to unlock the secret exit of K3.

Pirate

Pirate can be unlocked by clearing P3.

T-Rex

T-Rex can be unlocked by clearing C1.

Manly Rick

Manly Rick can be unlocked by clearing U4.

Fighter

Fighter can be unlocked by clearing K3.

Prototype

Prototype can be unlocked by clearing X3, AKA the end of the world map!

Octodad
Octodad can be unlocked by clearing N1.

Cogs Rocket
Cogs Rocket can be unlocked by clearing U5.

Meat Boy
Meat Boy can be unlocked by clearing K1.

Minecraft

And Minecraft can be unlocked by clearing F1!

Just a final little note to say thanks for reading my guide, and I hope you found it helpful! This is my first guide, so if I got anything wrong or anything bothers you about it, feel free to write a comment and I shall see if I fix it. I also want to thank the developers for making a great little game, and I hope their next game sells better than this game does!