There are a couple of strategies for understanding the real numbers. One is to write down a definition of real numbers, for example using rational numbers and Dedekind cuts, hoping that what you're describing is really what you mean. The other is to write down the properties of real numbers as you understand them as "axioms", and go from there. An important property of real numbers that always comes up (either as a consequence of Dedekind cuts or as an axiom itself) is the least upper bound property -- every set which has an upper bound has a least upper bound. That's what gives you the "completeness" of the real numbers, from which you can prove facts like the completeness of the real numbers (i.e., Cauchy sequences always converge), the Heine-Borel theorem (closed and bounded subsets of the reals are "compact", and vice-versa), and Cantor's intersection theorem (that the nested intersection of a sequence of non-empty compact sets is also compact).

The diagonalization argument is an intuitive tool, IMHO. It is great if it convinces you, but it's difficult to make rigorous in a way that everyone accepts due to the use of a decimal expansion for every real number. One way to avoid that is to prove a little fact: the union of a finite number of intervals can be written as the finite union of disjoint intervals, and that the total length of those intervals is at most the total length of the original intervals. (Prove it by induction.)

THEOREM: [0, 1] is uncountable. Proof: By way of contradiction, let f be the surjection that shows [0, 1] is countable. Let U_i be the interval of length 1/2*i centered on f(i). The union V_n = U_1 + U_2 + ... + U_n has combined length 1 - 1/2*n < 1, so it can't contain [0, 1]. Another way to state that is that K_n = [0, 1] - V_n is non-empty. K_n also compact, as it's closed (complement of V_n) and bounded (subset of [0, 1]). By Cantor's intersection theorem, there is some x in all K_n, which means it's in [0,1] but none of the U_i; in particular, it can't be f(i) for any i. That contradicts our assumption that f is surjective.

Through the right lens, this is precisely the idea of the diagonalization argument, with our intervals of length 2*-n (centered at points in the sequence) replacing intervals replacing intervals of length 10*-n (not centered at points in the sequence) implicit in the "diagonal" construction.