Mastery comes from problem solving and practice - not from reading books. So, I would advise students to limit what you read, but spend a lot of time in problem solving. Get the basics in place. Start with euclid's elements and master that first.
I think Euclid is fine as a historical document and interesting in a broad sense, but its kind of silly to start with a document from 300 BCE when you could start with, for example, "How to Solve It" (Polya). And even that text could use a rewrite to make it much more readable.
I have completed substantial education in both mathematics and physics and I would say one of the weaknesses of the standard system of courses in physics is that it more or less recapitulates the development of physics in a historical shape, which substantially obscures mathematical structures which are shared between disciplines developed at different times. For example, unless you were lucky or very curious you might never appreciate that the bras and covectors relate to kets and vectors in fundamentally the same way. You might only have had a vague sense that physics involves making a lot of sandwiches.
Don't get me wrong, I'd be delighted if my kid's school broke out The Elements, but I just don't think its an obvious pedagogical strategy to start math instruction (self or otherwise) there.
I meant that the foundations of what it means to do math can be found in as elementary a text as the Elements. Most importantly it shows how an abstract world (what we feel as space around us) can be modeled as Geometry with a set of axioms, along with a way to demonstrate statements without doubt. Just understanding this very deeply (in your bones) with a lot of problems would make you a better mathematician than if you read all the books listed above, at least in my view. Other areas of mathematics model different kinds of abstract worlds, but the activity is largely similar.
The problem I see is the focus on gathering knowledge without properly assimilating it (ie reading a lot of books). By assimilating I mean, being able to explain how something is constructed from the very basic first principles of how the conceptual structure was built.