There are a series of goals that you have to achieve to complete the game and unlock mechanics. In order, they are:

5 Attempts - Unlocks Distance

500 Distance - Unlocks Medals and Time

15 Medals - Unlocks Levels

10 Levels - Unlocks Coins

255 Coins - Unlocks Trophies

4 Trophies - Unlocks Zeniths

1 Zenith - Unlocks Nothing

2 Zeniths - Unlocks Nothing

3 Zeniths - Unlocks Nothing

4 Zeniths - Unlocks Green Coins

5 Zeniths - Unlocks Nothing

6 Zeniths - Unlocks Nothing

7 Zeniths - Unlocks "???"

??? - Unlocks Ending

To unlock ???, you have to wrap around the left and right sides of the screen in a way to make the white text read LRLLRLRR.
To finish the game at the ending, you have to explode on the right glass pane and go downstairs

There are 4 categories of medals in the game. Attempts, Distance, Time, and Medals.
These categories are fairly self explanatory, but when you'll complete them isn't.

The zenith for Attempts is at 2047
The zenith for Distance is at 32767
The zenith for Time is at 4095
The zenith for Medals is when you have gotten the zenith for every other category of medals

I'm not doing details as to the specifics of when each type of medal is unlocked.

There are a total of 8 trophies in Don't Move. They are as follows (in order):

Left Trophy: Die only on the left side for a long enough amount of time to get this trophy

Circle Trophy: Alternate between dying on the left and right sides of the screen long enough to get this trophy

Right Trophy: Die only on the right side for a long enough amount of time to get this trophy

Ninja Trophy: Stand still for a long enough amount of time to get this trophy

Fire Trophy: Stand still next to a torch long enough to get this trophy

Two-Way Trophy: Hold both arrow keys at once

Coin Trophy: Get 2 coins at once

Cotton Trophy: Get all other trophies

Coins and XP/Levels are gained passively when just walking around.

XP is gained more than one way. Its gained every time you do anything nearly. The Zenith is at 63 levels

Coins are spawned randomly throughout the hallway you are in and, after unlocking a certain number of Zeniths, are turned green. Green coins give a significant amount more coins than yellow coins and makes the Coin zenith actually possible. Speaking of, the Zenith is at 1,048,575 coins

There are a few extra secrets this game holds.
If you manage to reach the two middlest torches without dying, Jim (ghost ninja) will appear and fly across the screen.
Sometimes there will be a flower in the hallway. Just randomly.
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There are three stars that speed up the game. The codes are never shown in any degree, but still exist.
These are the codes. They each grant one star.
LRRLRRLRRLRRRLLRLLRLLRRR
LRLRLLRRLLRRLLLRRRLLLRRR
LLRRLRLRLLRRLRLRLLRRRRLL

If you enter the code below in the start screen before the screen fades to black, you will be given all three stars. This requires having reached the GAME OVER screen atleast once.
LLRRLRLRLRRR

Side note: There is a binary version of the konami code that you can input on the title screen when it fades to black to immediately unlock all of the stars, but it hasn't worked any time I've tried. If you can get it to work, comment please! The code: LLRRLRLRLRR