At least as a counter-point to the other good answers, I must confess that I have misgivings about the standard undergrad math curriculum in the U.S., primarily because I think it presents the subject as a plodding, exaggeratedly fastidious version of the most elementary parts of it... pointedly ignoring genuine motivations, historical motivations, chronology, and too many other aspects, while not allowing itself to mention any substantial contemporary developments.
That is, most often, the "logical order" precept has taken over, and students are taught to worry endlessly over potential pathologies.
At another extreme, there was an older tradition of "engineering math", which went too far in the opposite direction, but did have the virtue of addressing interesting questions, both internal to mathematics and external. Oddly, with the advent of good "math software", it appears that much of what was done by hand by engineers a few decades ago is essentially programmable...
Yes, I am proposing that "rigorous proof" is not the highest goal in mathematics, although, of course, knowing with greater certainty is desirable. Making implicit assumptions overt is good methodology. But being paranoid about technicalities is not good methodology.
A riff I use to explain to the best undergrads I see is that undergrad classes are essentially worthless as preparation for grad school, and the first-year grad courses are scarcely better. That is, quasi-ironically, "we" train beginners into a behavior pattern that we often seemingly endorse for them, but which is not at all what we do.
That is, the possibility of "logical perfection" in mathematics, unlike most other venues, inordinately catches our collective fancies, and become a goal in its own right. (And, possibly, it is worthwhile, but it is not the whole.)
Thus, my caution would be to disbelieve the worldview implicit in undergrad math curricula, and to distrust the choices of focus of it. Better if one can survive in relatively gentle graduate courses, even if they have degenerated (as is often so) into laundry-lists of must-mention topics.
In other words: mathematics is far simpler than as portrayed in undergrad courses.
Best to become acquainted with basics privately and efficiently, unless one truly needs the carrot-and-stick thing.