You get three candy for catching a Pokemon in Pokemon Go and one candy for transferring the Pokemon to the professor (or “meat grinder” to get 1 Soylent Green, as I like to call it). To get candy form the professor, click on the Pokemon then scroll (swipe up) to get to the bottom of the screen and click “Transfer”.
Correct me if I’m wrong but you also get 25 candy for hatching an egg that requires you to walk 10km, plus 1 if you send the Pokemon to the meat grinder (total 11 candy for a 10km egg). I don’t know how much candy you get from the other eggs.
Candy can only be used on the type of Pokemon it came with, or that Pokemon’s evolved / pre-evolved versions.