Due to the sheer number of spells available in the game, it's more than likely that many of these achievements can be earned in multiple ways. All this guide offers is at least one way to earn each of them, with a focus on simplicity wherever possible.

Q: Can all of these achievements be earned in one playthrough?
A: It will take at least two playthroughs due to exams 2 and 3 having two mutually exclusive achievements.

N.B: As the exact details regarding spell levels aren't online yet (or at least, I haven't seen them yet), the requirements for learning combination spells may be slightly lower than the ones given here.

Up and Out
"You exited the first exam by climbing the vines."

Requirements: Green Magic 10

Earning the Achievement: Navigate your way to the square with the vines. Once there, cast Heal (Green 10, MP cost 10) on the vines, then examine them to climb them out of the dungeon. You should earn the achievement upon exiting the dungeon.

Total MP cost: 15, assuming you used Light to light up your path.

Gas Safety
"You got through the second exam by turning off the gas."

Requirements: Red Magic 5

Earning the Achievement: Use the spell Push (Red 5, MP cost 5) to push the boulder in the top left part of the area into the square directly south-west of the first square of gas. This should take exactly three castings of Push and will allow you to travel safely to the stairs.

Total MP cost: 15

Left The Gas On
"You made it through the second exam with the gas still active."

Requirements: Blue Magic 30 OR Strength 41+

Earning the Achievement:
Option 1: Take the rightmost path to walk over to the gas. Use Farsight (Blue 15, MP cost 2) on the gas square farthest from you, then use Teleport Self (Blue 30, MP cost 20) to warp past the gas (and directly onto the stairs if you feel like it.)
Option 2: Simply walk straight through the gas, tanking the damage with your massive strength. You'll earn no merits, but you will earn another achievement in the process (see section You're Doing It Wrong for more information.)

Total MP cost: 22 for option 1, 0 for option 2

Shortcut
"You traveled through the third exam using the rune circle."

Requirements: White Magic 20 OR Black Magic 10

Earning the Achievement: To start with, you must find the false wall along the length of corridor you begin in. This can be done with Awareness (White 15, MP cost 10), True Sight (Blue 10, MP cost 2). or by casting Inspect (Black 1, MP cost 1) on each of the walls till you hit the jackpot. Once you find the rune circle behind the false wall, cast Spirit Echoes (White 20, MP cost 5), or Trigger (Black Magic 10, MP cost 5) and then examine the circle to touch the correct runes. Once on the other side, make your way out with the aid of another Awareness or any cantrip that strikes your fancy.

N.B: The rune circle you need to activate is not the one you find on the floor past the pits. The circle you need to activate is hidden from view along the southern wall near where you begin and must be found using one of the above methods.

Total MP cost: 25, assuming the MP-intensive Awareness route. Sub-10 is eminently possible.

Move Along Home
"You crossed directly over a hole to defeat the third exam."

Requirements: Red Magic 25, Green Magic 20

Earning the Achievement: Moving onto our first dichromatic spell... take the rightmost past till you reach the pit, then cast Long Jump (Red 25, Green 20, MP cost 15) to leap past it. Make sure to cast Light (Blue 1, MP cost 1) on the darkened square you land in, then make your way out of the dungeon as usual.

Total MP cost: ~26. sub-20 is perfectly doable.

Lightning Dodge
"You defeated the fifth exam without either disabling or being hit by the trap."

Requirements: Blue Magic 20, Black Magic 18. I also recommend getting White Magic 10 and something that will allow you to cross the lava, such as Teleport Self (Blue 30, MP cost 20) or Long Jump (Red 25, Green 20, MP cost 15)

Earning the achievement: To trigger the trap without getting hit, cast Televoke (Blue 20, Black 18, MP cost 10) on the pedestal while standing away from it. To make your life easier, cast Communication (White 10, MP cost 5) on the Ghost Wolf when you meet it and make sure you have the means to get to and away from the pedestal.

Total MP cost: 55, assuming the costliest route of Communication + Televoke + two castings of Teleport Self. You can shave the number down by wearing a Blue Charm (total MP cost: 30) or by experimenting with other spells.

Let Sleeping Moths Lie
"You won the exam without waking the mothmen."

Requirements: Blue Magic 5. Anything that will let you pass safely through the hazards (such as Blue Magic 30 + White Magic 15)

Earning the Achievement: Make your way to where the poor munchkin is being held. Before freeing it, cast Silence (Blue 5, MP cost 2) on the gate. Then simply wait and enjoy your new achievement.

Total MP cost: 33, assuming ye olde Awareness + Teleport Self combo. Can be shaved down to 22 with a Blue Charm, and likely even further down with a different way through the hazards.

Big Jelly Loves You
"You won the final exam without killing the Big Jelly."

Requirements: Green Magic 32, White Magic 35. Smarts 35+, the more the safer, or alternatively a Green/White Charm.

Earning the Achievement: seek out the Big Jelly (for instance, by using Luke's Track Scent (Green 1, MP cost 1)) and make sure you can target it (e.g. by using Donald's True Sight (Blue 10, MP cost 2). That done, simply cast Sleep (Green Magic 32, White Magic 35, MP cost 25) on your gelatinous friend. Take the orb and make your way to the exit. Remember that everyone in the party needs to have 10 MP left to use the apparatus at the end.

Total MP cost: 35, assuming no charms and the PC using no spells besides Sleep while also not getting hit by the Big Jelly's effect. A charm and/or more MP is highly recommended.

You're Doing It Wrong
"You defeated a dungeon exam without casting any spells."

Requirements: As the achievement description states, you must pass an exam without using any magic. I believe that with sufficient strength, this can be accomplished with the majority of exams.

Earning the Achievement: There are multiple ways to do this, so I'm offering two simple ones.
Option 1: before Exam 2 (October 25), make sure you have at least 41 Strength. Simply walk right through the gas to the exit.
Option 2: during Exam 6 (February 21), find the gate, wait for the frog to open it, and rush forward to pinch the chalice.

Total MP cost: If you used any MP, you failed.

If you know the more exact requirements for learning combination spells, have found easier ways to earn the achievements above, or simply wish to share some feedback, don't hesitate to hit me up. Happy dungeon crawling!

Special thanks to Nemo2342 for the Trigger solution to the Shortcut achievement.