ARC welder is really just a stick welder or at least thats the way we always referred to them, and the other unit we have is a wire welder - wire on a spool...
Nestor covered it pretty well... heres some other thoughts.
MIG (Wire) Welding Benefits
* Easiest to learn
* Better control on thin metals
* Out-of-position welding
* Deep penetration for welding thick sections
* Same power source for solid & flux cored (gasless) wire
Stick Welding Benefits
* Skill level required: Moderate
* Suited for windy, outdoor conditions
* Forgiving on dirty or rusty metal
* Can weld at a long distance from machine
I learned how to weld through trial and error, with both stick and wire - we always built our own equipment and building - so the need was there for me to learn.... I always loved my dads approach to teaching, he gave me some stuff to weld, and after I finished he would come over to take a look and then he hit it as hard as he could with a big hammer ( HEY !!!!) - if it broke, he said you did it wrong, smiled, said do it again then walked away
I got the hang of it after awhile.... LOL
Take a look here for some good articles of info... Miller has a vast resources and learning area worth spending some time at...
http://www.millerwelds.com/resources/improving-your-skills/