Perhaps this was merely an impression of yours, but it's hard to say what is it that concerns you. What subjects, in particular, you found in difficulty more often than your friends? Ask yourself this question first.
Your friends probably didn't have a "deeper" understanding of logic than you, but they could have had a more intuitive understanding of the subjects. Which is to say, for example, that they had a easier time following the steps of the a mathematical demonstration. This wasn't necessarily due to a familiarity with "logic", which is only a narrow field of knowledge, but rather their familiarity with definitions necessary to make such demonstrations, so that in their heads they were really "following" the teacher, rather than only being shown the steps.
Now, though I've said "logic" is only a particular field of knowledge, since you seem to be concerned with reasoning in general, there's no reason to avoid it. In fact, it might greatly beneficial. At a layman's level, Wikipedia suffices.
You could also try to dabble in philosophy, and to try to exercise your thought by understanding arguments. Platonic dialogues, though rather dull, portray philosophy in a very natural language, and characters present their viewpoints all in the form of arguments.
But really, anything you choose, so long as it's intellectually challenging, will be to your benefit.