Note that a fallacy is a reasoning flaw; sometimes the goal might be to trick you, indeed. But sometimes it's just a brainfart... or you might be dealing with something worse, like sheer irrationality. That said:

• look for the conclusion. What is the point that the writer is delivering? (Note: you might find multiple conclusions. That's OK.)
• look at what's being used to support that conclusion. What is the core argument?
• look for the arguments used to feed premises into the core argument. Which are they?

Then try to formalise the arguments that you found into "premise 1, premise 2, conclusion" in your head or in a text editor. Are the premises solid? Do you actually agree with them? And do they actually lead into the conclusion? If something smells fishy, you probably got a fallacy.

Get used to at least a few "big" types of fallacies. There are lists across the internet, do read a few of them; you don't need to memorise names, just to understand what is wrong with that fallacious reasoning. This pic has a few of them, I think that it's good reference material, specially at the start:

In special I've noticed that a few types of fallacy are really common on the internet:

• genetic fallacy - claiming that an argument is true or false because of its origin. Includes ad hominem, appeal to nature, appeal to authority, ad populum, etc.
• red herring - bringing irrelevant shit up as if it supported the conclusion, when it doesn't matter. In special, I see appeal to emotion (claiming that something is false/true because it makes you feel really bad/good) all the time.
• oversimplification - disregarding key details that either stain the premises or show that they don't necessarily lead to conclusion. False dichotomy ("if X is true, Y is false" in situations where both can be true or false) is a specially common type of oversimplification.
• strawman - distortion of an opposing argument into a way that is easier to beat. Again, notice that "intention" doesn't matter; only that the opposing argument isn't being addressed.
• moving goalposts - when you counter an argument, the person plops another in its place, without acknowledging that it's a new argument. Often relies heavily on ad hoc (making stuff up on the spot to shield an argument)
• four terms - exploiting multiple meanings associated with the same word to create an argument like "A is B¹, B² is C, thus A is C".

There are also some "markers" that smell fallacy for me from a distance. You should not trust them (as they might be present where there's no fallacy, or they might be absent even when the associated fallacy pops up); however, if you find those you should look for the associated fallacy:

• "As a" at the start of a text - genetic fallacy, specially appeal to authority
• "Trust me" - red herring, specially appeal to emotion (once you contradict the argument there's a good chance that the other will create drama because you didn't blindly trust them, so the whole thing boils down to "accept this as true otherwise you'll hear my meltdown").
• "I don't understand" followed by a counter-argument - strawman. Specially common in Reddit.
• "Actually" - red herring through trivia that is completely irrelevant in the context.