One reason people use Spivak's book for self-study is that it has a very extensive solution manual containing the answers to the exercises, so if you are studying by yourself you can look up all the answers. Now there is stackexchange so the solutions manual is less relevant, although some people might be embarrassed to admit when they got stuck on question 1.1 or something. Spivak also grades the difficulty of the problems with stars, which are quite accurate about the relative amount of time the problem will take.
It was published in 1966 so it's already "traditional". Also Spivak was the course book for first-year analysis at my university, so I really don't think it's some kind of unconventionial book (or perhaps I went to a weirdo university). Anyway I just went and looked at the university's page and they are still using Spivak. Spivak was ahead of his time because he also invented "gender-neutral pronouns".
If you do decide to go with Spivak it would be smart to start from the beginning and do the "easy" problems and then work through to the end. People who start in the middle of the book end up like those weightlifters who only work on one part of the body, so they have giant biceps and tiny legs or something. Then they can do all kinds of complicated-looking proofs but they cannot do the simple parts of problems.
Spivak also does not cover set theory. Sometimes you need to understand at least the basics of that.
Another opinion: When I mentioned that Spivak was the course book at my university, a person who attended another place told me that he thought that "Spivak has no soul". I have absolutely no idea what he meant by that comment.