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In addition to the answers that have already been given, I think another reason that mathematics doesn't collapse is that the fundamental content of mathematics is ideas and understanding, not only proofs. If mathematics were done by computers that mindlessly searched for theorems and proof but sometimes made mistakes in their proofs, then I expect that it would collapse. But usually when a human mathematician proves a theorem, they do it by achieving some new understanding or idea, and usually that idea is "correct" even if the first proof given involving it is not.

One recent and well-publicized story is that told by the late Vladimir Voevodsky in his note The Origins and Motivations of Univalent Foundations. Here's a bit of one story that he tells about his own experience:

my paper "Cohomological Theory of Presheaves with Transfers," ... was written... in 1992-93. [Only] In 1999-2000... did I discover that the proof of a key lemma in my paper contained a mistake and that the lemma, as stated, could not be salvaged. Fortunately, I was able to prove a weaker and more complicated lemma, which turned out to be sufficient for all applications....

This story got me scared. Starting from 1993, multiple groups of mathematicians studied my paper at seminars and used it in their work and none of them noticed the mistake.... A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.

I don't know any of the details of the mathematics in this story, but the fact that he was able to prove a "weaker and more complicated lemma which turned out to be sufficient for all applications" matches my own experience. For instance, while working on a recent project I discovered no fewer than nine mistaken theorem statements (not just mistakes in proofs of correct theorems) in published or almost-published literature, including several by well-known experts (and two by myself). However, in all nine cases it was simple to strengthen the hypothesis or weaken the conclusion in such a way as to make the theorem true, in a way that sufficed for all the applications I know of.

I would argue that this is because the mistaken statements were based on correct ideas, and the mistakes were simply in making those ideas precise. Or to put it differently, mathematicians get our intuitions from "well-behaved" objects: sometimes that intuition can be wrong for "pathological" objects we didn't know about, but in such cases we simply alter the definitions to exclude the pathological ones from consideration.

On the other hand, people do sometimes get mistaken ideas. For instance, here's another quote from Voevodsky's article:

In October 1998, Carlos Simpson ... claimed to provide an argument that implied that the main result of the "∞-groupoids" paper, which Kapranov and I had published in 1989, cannot be true. However, Kapranov and I had considered a similar critique ourselves and had convinced each other that it did not apply. I was sure that we were right until the fall of 2013 (!!).

I can see two factors that contributed to this outrageous situation: Simpson claimed to have constructed a counterexample, but he was not able to show where the mistake was in our paper. Because of this, it was not clear whether we made a mistake somewhere in our paper or he made a mistake somewhere in his counterexample. Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson’s paper appeared, both Kapranov and I had strong reputations. Simpson’s paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it.

In this case I do know something about the mathematics involved, and my own opinion is somewhat different from Voevodsky's. In the 2000's I was a graduate student working on higher category theory, and my impression was that in the community of higher category theory it was taken for granted that Simpson's counterexample was correct and the Kapranov-Voevodsky paper was wrong, because the claimed KV result contradicted well-known ideas in the field.

The point here is that a community of people developing ideas together is likely to have arrived at correct intuitions, and these intuitions can flag "suspicious" results and lead to increased scrutiny of them. That is, when looking for mistaken ideas (as opposed to technical slips), it makes sense to give differing amounts of scrutiny to different claims based on whether they accord with the intuitions and expectations of experience.

So what do you do as a student? In addition to the other good advice that's been given, I think one of your primary goals should be to train your own intuition. That way you will be better-able to evaluate whether a given result, or something like it, is probably true, before you decide whether to read and check the proof in detail.

Of course, there is also the position that Voevodsky was led to:

And I now do my mathematics with a proof assistant. I have a lot of wishes in terms of getting this proof assistant to work better, but at least I don’t have to go home and worry about having made a mistake in my work.

I have a lot of respect for that position; I do plenty of formalization in proof assistants myself, and am very supportive of it. But I don't think that mathematics would be in danger of collapse without formalization, and I feel free to also do plenty of mathematics that would be prohibitively time-consuming to formalize in present-day proof assistants.

1. Redundancy is one big source of self-healing. A result with three different proofs is rather unlikely to be wrong. Also, people try to apply fresh results; wrong results often lead to contradictions when applied, alerting mathematicians to their wrongness. Same for proofs: Mistakes in proofs are often spotted when someone tries to adapt the proof to other questions.




2. This is tricky. These days, using Google Scholar's "cited by" feature and various other backlink aggregators, you can get a list of papers/book that reference a given paper. Thus, if you find an error in the literature, you can track down where the "corruption" has spread. But getting corrections published is very difficult. Ted Hill and Nikolai Mnev are known for having struggled through the whole process of correcting someone else's false claims, but lots of people end up staying silent or (these days) just posting what they know somewhere on a forum like MathOverflow when someone stumbles upon the same problem. Then there are situations where no specific error can be pinpointed, but important material is simply imprecise and unreadable; fields often linger in such a limbo until someone does the thankless job of building the foundations underneath them. Katrin Wehrheim is one example of this.




3. This question of mine got 41 votes, so yes, this is a fairly well-acknowledged problem.




4. Ask your advisor and others. You definitely want to understand all proofs in undergraduate and lower-level graduate classes; they aren't particularly likely to be wrong, but you'll use the ideas anyway. As for advanced theory you rely upon, it depends.

This is a broad question, but you may find it helpful to read The Existential Risk of Math Errors. It suggests a certain robustness of the mathematical edifice, which I actually think extends to the natural sciences as a whole. (Newtonian mechanics is "wrong" in a fundamental sense, but neither the development of relativistic mechanics nor the discovery of quantum mechanics has caused the collapse of classical mechanics.)

This quote in particular from Gian-Carlo Rota bears on your points 1 and 2:

When the Germans were planning to publish Hilbert’s collected papers
and to present him with a set on the occasion of one of his later
birthdays, they realized that they could not publish the papers in
their original versions because they were full of errors, some of them
quite serious. Thereupon they hired a young unemployed mathematician,
Olga Taussky-Todd, to go over Hilbert’s papers and correct all
mistakes. Olga labored for three years; it turned out that all
mistakes could be corrected without any major changes in the statement
of the theorems. There was one exception, a paper Hilbert wrote in his
old age, which could not be fixed; it was a purported proof of the
continuum hypothesis, you will find it in a volume of the
Mathematische Annalen of the early thirties. At last, on Hilbert’s
birthday, a freshly printed set of Hilbert’s collected papers was
presented to the Geheimrat. Hilbert leafed through them carefully and
did not notice anything.

If a result is not used much, then its veracity does not matter much for the rest of mathematics.

Otherwise, there may be several proofs of the result, which makes it much more probable (in a general sense) that the result is true. Importantly, there is usually some explanation or understanding of why the result is true, that is, the ideas behind the result.

Also, people try and do construct counterexamples to disprove a result, if they do not see why the result should be true.

It might feel like the chain of citations goes ad infinitum,
but of course this can't literally be the case. Indeed as long as
mathematics is (at least primarily) an endeavor by and for humans,
the entire chain of reasoning must be graspable by about the time a mathematician
reaches PhD level. Much of it will not yet be accessible to an advanced
college student such as the OP, and modern mathematics has gone in
so many directions that no one mathematician can grasp more than
a small sliver of the frontier $-$ which is why modern research mathematics
requires specialization, and a mathematical community large enough to support
such a large frontier. Inevitably errors occur, and a few of those
propagate for some time before being caught. But the enterprise as a whole
is self-correcting (as already explained in several ways in other answers).

While the responses to this question have so far have generally (and correctly) focused on the robust nature of the edifice of mathematics as a whole, it might be worth pointing out by way of contrast that sometimes there are indeed doubts about foundational issues within a specific subfield (because, e.g., an "important" paper is known to have flaws), and that can be extremely deleterious to the field in question. Something along these lines is discussed in this Quanta magazine article: https://www.quantamagazine.org/the-fight-to-fix-symplectic-geometry-20170209/. (And I would say that what is described there is not even the most extreme example of what can happen.)

Further to M. Shulman’s advice to “develop your intuition”, it’s probably worth adding that this is often done by understanding many examples, special cases whose moving parts are already transparent to you. You get something simpler and more robust. And psychologically at least, “trust” in a result often relies on familiarity with a library of such cases, more than on line-checking a general proof. (I would also guess that many innocuous “errors” only reflect some overshoot in generality when streamlining things for publication, and that is why not everything collapses.) Or, in the polemical words of Arnol’d (2004):

There are two principal ways to formulate mathematical assertions (problems, conjectures, theorems,...): Russian and French. The Russian way is to choose the most simple and specific case (so that nobody could simplify the formulation preserving the main point). The French way is to generalize the statement as far as nobody could generalize it further.

(That’s not to say that “top-down” never wins — e.g. reputedly, only the discovery of Bott periodicity settled a “spirited controversy” on specific homotopy groups (1959, p. 355 and Math review).)

Well, because of a multitude of reasons, including:

1. Mathematicians rarely make mistakes in simple proofs. And a lot of the basics of math are simple theorems that follow easily from the formal definitions.


2. 
   

   Mistakes get corrected quickly before claims get accepted and used:

   
   
   

   2.1 Most people double-check and re-check their own proofs before claiming to have proved something significant (although some don't - I've heard a rumor that Saharon Shelah had a bunch of papers with errors and brushed this off by saying they could be be easily rectified)

   
   
   

   2.2 Mathematicians are a community and check each other's work - so there's much less chance of mistakes escaping scrutiny.

   
   



3. Doubt: People - mathematicians and users of math - don't just accept new claims at face value. Even if they don't have the time to check proofs themselves, they'll treat new claims/results as somewhat suspect until sufficient (or apparently sufficient) verification happens.




4. Using an erroneous theorem usually leads to obvious problems, so whoever adopts an erroneous theorem as valid typically falls flat on their face, and this easily leads to doubt being cast on their assumptions - namely, the faulty theorem.



5. 
   New mistakes don't invalidate past work. If a field of mathematics has some valid, no-mistakes basis, then our mistakes building on that basis don't invalidate it. At most, we may get confused and doubt some of it - but that will only make us double-check its validity or look for a counter-example - which will fail.

Sometimes these collapses happen. Ole Peters and Murray Gell-Mann have a 2016 paper about a mistake in Expected Utility Theory, which they track down to a mistake in a Bernoulli paper from 1738.

I'm an outsider looking in, but it sure looks like much of that entire field is in serious trouble. There's a post on this here: https://ergodicityeconomics.com/2018/02/16/the-trouble-with-bernoulli-1738/