I think there are two aspects to logic which are more important above all others. The first is to understand what makes a valid logical argument. At the moment, mathematics' Proof Theory is probably the most extreme version of that, but you don't need to go that far. The key to this is being able to say "If I start with a set of statements that represent things that mean something true (semantic truth), then here is a set of operations I can do on those statements, regardless of what they truly mean, which result in the production of additional true statements (syntactic manipulation).
To give an example, proof theory will dig into your "2+2 = 4" because it is empirically true that if I take two pencils, and gather two more pencils, the description of that is 4, and argue that "2+2=4" is true regardless of the empirical side. In particular, we typically do not define 4 to be the result of adding 2 + 2. It is usually defined to be the number following the number following the number following the number following the number zero (Peano arithmetic people would write that as
Su(Su(Su(Su(0)))) where Su is the "successor function"). Two would be defined as the number following the number zero (written
Su(Su(0))). It can then show that 2+2=4 using the rules of Peano arithmetic, which are all manipulations of symbols (such as
0 + a =a and
a + Su(b) = Su(a) + b). We also use logic to show that this addition operator has consistent properties like the ones we are used to (such as
a + b = b + a).
Note that that entire section was devoted to manipulation of the abstract symbols, without mention of the meaning behind those symbols. That's one half of the story. The other half of the story is being able to determine valid abstract symbols to represent the real life events that you are talking about. Mathematics has Model Theory, which is very good at handling this once you've already abstracted things to mathematical symbols, but philosophy opens the door for many opinion about what abstract symbols and what rules are "valid."
In your example, we can say "If I take two pencils, and gather two more pencils, then the number of pencils I have is well described using 2 + 2 = 4." How do I prove that? I have to convince you that pencils can be represented by numbers, and thus the laws of addition apply to them.
Eventually, I'd recommend everyone read about Tarksi's Semantic Theory of Truth. It's a rather quirky little beast, which you may not agree with, but which is very good for demonstrating the nature of this process of converting meaning into symbols so that an argument can be made.
Finally, I'd pick a few famous systems of logic. A logical proof is only valid if the person admitting the argument believes that kind of logic is true. For example, an argument involving infinity would be rejected by a finitist. I would argue the most influential logic in the Western world is propositional logic, so no matter what you do, it would be worth your time to be familiar with its rules and patterns. Beyond that, it's up to you. Myself, I find First Order Logic to be useful, despite its flaws, but the systems you research are really up to you.