Pointing out logical fallacies is pointless. Though it is useful to study logical fallacies so that you can analyze your own thinking and make adjustments, but to point out the logical fallacies in someone else's thinking is entirely counter productive.
If you are in a disagreement with someone, by definition they do no view you as an authority on the subject and thus they are unlikely to accept your analysis of whether or not their thought process is fallacious or not. The reason why they think the thoughts they do is because they think their thought process is right, and no one likes to be told they are wrong.
The whole reason why they are disagreeing with you in the first place is because they think your thought process is out of order. So if you attempt to point out an error in their thinking they are not disposed to begin an introspective analysis of their thinking. Almost by definition, your thinking is in error, so any attempt to point out an error in their thinking will be rejected immediately. Thus it is pointless and counter productive to point out the fallacies in someone else's thinking.
This does not mean that you never point out logical fallacies, just do not expect the person you are critiquing to respond positively. It is also justified for certain egregious misuses of logic and reasoning, but not as a tool to convince those afflicted, but to warn those who are unconvinced and can be swayed by your reason, that is, those who have not taken a position yet.
But it is a very good idea to study logical fallacies so that you can improve your own thinking. A good place to start is by analyzing what other people say and see if you can pick out the logical fallacies. First start with arguments you disagree with. Those are the easiest. Next you move to arguments you have no opinion about one way or the other. This allows you to dispassionately assess the arguments. When you are comfortable with those you can move on to arguments that you naturally agree with. Those are the hardest arguments to analyze, with the exception of your own arguments. It takes a lot of humility to analyze your own thinking for fallacies.
Some good fallacies to start with are perhaps the most famous ones such as ad hominem, slippery slope, or the post hoc fallacy. Once you can identify fallacies in arguments that you already disagree with, then you can try moving on to something more difficult. Just remember that pointing out a fallacy in someone else's argument is (almost) never productive. I can think of one, and only one, case when I pointed out someone's fallacious thinking and it actually made an impact on them, and they admitted that I "got them" and for a moment their defenses were down and I was talking to a human being. Unfortunately it did not last.
Remember the point of studying fallacies is to eventually analyze your own thoughts and learn to remove fallacious thinking. When you do that then you can begin to build convincing arguments that can sway people to your position.