PROOFS. Which one, when.

Direct Proof.
Proving the Contrapositive.
Proof by Counterexample.
Proof by Contradiction.

How do you differentiate when to use each of these?

Depends really.

I tend to use counterexamples a lot more for analysis type things, because there's often some nice pathologic examples that work for a lot of things, for example if you're talking about Banach spaces then l_inf or l_0 are often great for counterexamples because they're not reflexive, or because l_inf isn't seperable etc. If you do enough questions you tend to see which ones are good to use, spike functions etc.

If you know the proof's going to be just basic manipulation and applying definitions, but it might get complicated, then it's generally a toss up between direct proof and contrapositive. One's going to be easier, but it's not normall extremely obvious to tell.

All I can say is, do lots of proofs, you'll get a feel for it.