I'd like to use the arduino uno, an air pump, a 12 relay, and something to release pressure at X amount to throw darts at a dartboard with a velocity of 4m/s or whatever works. Any tips?
How about a crossbow?
Use an air piston to pull back the string and some kind of quick release to fire it. Arduino will control the air pump which moves the piston, the relay switches on/off or to pull and contract the piston.
A simpler alternative is a blow dart gun. But then what would you do about the dart's fin?
Depends what country you are in.
In uk crossobow ok.
Pneumatic you limited to 12 ft lbs energy without a firarm licence.
Pneumatic tennis practice machine illegal.
Tyre propelled type ok even if higher energy.
Dear God, I'm not trying to kill something, just throw darts. I crossbow is too much. I would like a reloadable blowgun or something that could really shot a throwing dart.
Ok then how about something like a football launcher? Put the dart on a track and push it between two fast spinning wheels.
I still don't know about the darts fin, it will most certainly get crushed between the wheels.
If you need the fin on when the dart is shot, then a crossbow like rig is the way to go. Note: you can use a regular rubber band, and not an actual crossbow string 
You might modify a paint ball gun to shoot darts, but auto loading the darts might be an issue.
4 m/s? That's not really very high speed.
In any case, the method of propulsion is dependent on what kind of darts you want to throw, how many of them, how often etc...
Local laws and regulations put aside (you're the one that should worry about staying inside them) pneumatic propulsion is feasible. Are you familiar with APFSDS (armour-piercing fin-stabilized discarding-sabot) ammunition? It's a type of ammunition modern tanks use. The business bit (projectile) is of smaller caliber than the barrel of the gun it is shot from. The gap is filled with lightweight material that is discarded when the projectile leaves the barrel.
This approach would allow you to have a two stage pneumatic system. You keep your main air supply in a large, high pressure tank and transfer a correct amount of air to the smaller tank. Using a variable pressure valve you can decide how much air you transfer to the smaller tank. When you fire your device, the air is released from the smaller tank and you should get a fairly good consistency.
This is pretty much how modern, regulated PCP (pre-charged pneumatic) air rifles work.
However, my concern is that your target 4 m/s speed is too slow for this method.
Interesting experiment. Some random ideas:
Use a 1/2 or 1/4 piece of tube with a slot for a vertical driving pin/arm. This arm is connected to a piston connected via solenoid to pressure source. Or you could use a straight push rod connected to a piston to push the dart down the half tube. For better accuracy a clamp assembly to hold the dart in position while being accelerated would be nice but mechanically intricate (should hold dart firmly during acceleration but then release cleanly without interfering with the fins). You could control firing velocity by either controlling the open time of the solenoid valve or by controlling the source pressure (I think this may be more repeatable but is probably more complex to implement).
Install on pan/tilt bracket to aim. Add camera with image processing to identify dartboard + algorithm to determine distance based on diameter of largest circle/oval detected (hopefully it is really a dartboard, else install safety mechanism that locks out firing if the camera doesn't detect dartboard).
Perhaps also use camera to track darts and write training algorithm for autocalibration. This bit may tax the ATmega based Arduinos a bit, but what fun is a project without challenges?
encryptor:
Dear God, I'm not trying to kill something, just throw darts. I crossbow is too much. I would like a reloadable blowgun or something that could really shot a throwing dart.
No quite , but darts are heavy and its quite easy to exceed th 12 ft lb limit.
Iknow of someone who got "done" for trying to build a clay pigeon launcher using compressed air.
It was elegant compared to the current motor spring method, unfortunatley under british law its an offensive weapon.
Im pretty sure though that using compressed air to cock a spring is ok.
I like the sabot idea, but you'd have to be very careful to ensure a symmetrical separation.
( I think there was once a very similar Scrapheap Challenge solution,, but I can't quite recall the problem - repeating crossbow?)
Golfball gattling gun ?
Its possible to get paintball miniguns terminator style, maybe that would give some ideas.
Nerf dart chaingun
ok like 80mph speed! lol! I was just guessing with the 4m/s. I just remembered from Physics that velocity had that unit.