I'm asking this as an autodidact who wants to learn math rigorously for its own sake. And I was just wondering if understanding proofs could be achieved without a formal grounding in symbolic logic. I ask because I have all the books I need but I simply don't have the patience I'd like to have for the formal logic book, as I'm itching to get into continuous math.
And to clarify, I'm interested predominantly in calculus/analysis, ODEs/PDES, and differential geometry.
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In order to study analysis and related fields you don't need any formal training in logic. At the same time, to learn computer science you don't need any formal training in analysis. Yet most universities require their students to study analysis to some degree. Why?
The reason is not to torture the students, or cull the first year acceptance rate. Or at least ideally these are not the reasons.
When you study abstract mathematics, you learn how to analyze a problem, and how to solve a problem using abstract thinking. The same principle can be applied to anything. Studying mathematics is the study of how to analyze, generalize and solve problems. And the more ways you know, the better you will be.
So while you don't need any formal training in logic and set theory, I would very much recommend at least a very good understanding of the basics of these fields. The basics of propositional calculus, predicate calculus, and naive set theory. These tools are very useful to mathematics, even if you don't apply them directly. They allow you to access better and higher understanding of the problems that you deal with, and how to deal with them.
As for my learned colleague who said that "this is basically common sense", while this is not far from the truth, you'd be surprised how many students I have seen having trouble with understanding the importance of quantifier order, or how to negate a proposition (with or without quantifiers). And Certainly understanding what does it mean for a function to be "not continuous everywhere" is important if you want to do analysis. (And you'd be surprised how many students will not be able to write that statement correctly.)
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Nonstandard analysis consists of one branch of formal logic. The Wolfram page in particular says:
"Crucially, however, the angle at which the nonstandard analyst looks at the axioms of analysis provides for an average case reduction in complexity that provides shorter proofs of various results, and will one day lead to the proof of a result which is not accessible to classical mathematics without nonstandard methods, precisely because its classical proof is too long to write down in the length of time humans will reside on Earth."
So, sure you can learn some of continuous mathematics without any formal logic. However, for a deep comprehension of the subject, learning some formal logic will probably serve you well.
Certainly no deep knowledge is required, even for subjects where foundational issues come up. Some knowledge about quantifiers is at times useful, but not really necessary, since lots of people consider the use of quantifiers to be bad style (otherwise they would be used lots in serious books, but they are not); these things have no content (merely different language). This holds for all branches of mathematics apart from logic itself, for the things which you are interested in in particular. You can easily read standard PDE books such as Trudinger/Gilbarg without knowing hardly any formal logic at all (with only a working knowledge), in fact I do so and I am certainly not the only one. But again, this is in a sense not surprising. I mean of course you should know the difference between $\forall x \exists y$ and $\exists x\forall y$, but this is basically common sense. Probably you don't need any formal logic at all.