mathematical physics - Applications of the Spectral Theorem to Quantum Mechanics

I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum Mechanics.

But let me first state the formulation of the theorem that I'm using:

Let $H$ be a Hilbert space. There's a 1-to-1-correspondence between self-adjoint operators $A$ on $H$ and spectral measures $P^{A}$ given by $$A~=~\int_{\mathbb{R}} \lambda ~dP^{A}.$$ ($\lambda$ denotes a constant, $\mathbb{R}$ denotes the real numbers.)

A corollary is:

Let $g:\mathbb{R}\to\mathbb{R}$ be a function. (Again: $\mathbb{R}$ denotes the set of real numbers.) Then: $$g(A)~:=~\int_{\mathbb{R}} g(\lambda)~ dP^{g(A)}$$
$$P^{g(A)}(\Delta) ~=~ P^{A}(g^{-1}(\Delta))$$ where $\Delta$ denotes a set in the $\sigma$-algebra of $\mathbb{R}$.

Okay. Now this is the theorem. First I don't really the application of the corollary in Quantum mechanics. I've heard that suppose you're given an operator $A$ this means that it's easy for you to define operators like $\exp(A)$, especially on infinite dimensional Hilbert spaces. This indeed could be useful in quantum mechanics. Especially when thinking about the "time-evolution operator" of a system.

However then I say: Why do you make things so complicated? Suppose you want to calculate $\exp(A)$. Why don't you define $$\exp(A)~:=~1+A+1/2 A^2 + \ldots$$ and require convergence with respect to the operator norm. An example: Consider the vectorspace spanned by the monomials $1,x,x^2,\ldots$ and let $A=d/dx$. Then you can perfectly define

$$\exp(d/dx)~:=~1+ d/dx + 1/2 d^2/dx^2 + \ldots$$

and require convergence with respect to the operator norm.

In addition to that I've heard that the spectral theorem gives a full description of all self-adjoint operators. Now why is that the case? I mean okay..there's a one to one correspondence between self-adjoint operators and spectral measures...but why does this give me any information about "the inner structure of the operator"? (And why is there this $\lambda$ in the integral? Looks somehow like an eigenvalue of $A$? But I'm just guessing)

I'd me more than happy, if you could provide me with some intuition and ideas of how the theorem can be used.