I'm studying Bernard Aupetit: A Primer on Spectral Theory
but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things?
Quite expansive and well detailed are the two books :
Dunford & Schwartz, Linear Operators, Wiley Classics Library, 1971
Rudin, Functional Analysis, McGraw-Hill, 1991
Less specialized but treating very well some parts of the topic is the Brezis, Functional Analysis too, and there are many exercises.
Hoping it will be of some use.
As nobody has mentioned Reed and Simon yet, I will do so:
Reed, Michael and Simon, Barry. Methods of Modern Mathematical Physics, Academic Press, 1980.
Of course "spectral theory" means different things to different people, depending on what they plan on doing with it. As the title suggests, Reed and Simon is in principle aimed at mathematical physicists (quantum mechanics, etc) but it is an honest mathematics textbook (all theorems are proved, etc). The first volume begins with the basics of functional analysis and ends with the spectral theorem, and volumes 2-4 proceed from there. I think it's a good treatment for any "working analyst"; it's well motivated and down to earth. Applications to topics such as PDE are made more explicit than in texts such as Conway. There are also a ton of really good exercises.
This book is a little unusual (in what I think is a good way) in that it includes, and shows the benefits of, Halmos's "multiplication operator" version of the spectral theorem, mentioned already by Jon Bannon.
Unfortunately the current edition is very expensive, so you may want to try to borrow it from a library or colleague at first.
I'm a big fan of the exposition in J. M. G. Fell & R. S. Doran, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles:
You might not like that it goes "from the abstract to the concrete", but as such it's really beautifully done. There are quite a few examples although it doesn't treat essential self-adjointness, deficiency indices, or the self-adjoint extensions of Laplacian plus potential type operators. For that, there is always Riesz-Nagy.
I very much like the treatment in "A Course in Functional Analysis" by John B. Conway. Also, there is a nice historical account of the Spectral Theorem in the American Mathematical Monthly by Lynn Steen, available here .
There is a short (121 pages) and clear exposition by a specialist : "Notes on Spectral Theory", Sterling K.Berberian (D.Van Nostrand Company,Inc.,1966).
There are excellent suggestions above. Let me add a footnote (since I am not suggesting a book).
Halmos's viewpoint on this is excellent for getting your feet under you (this was introduced to me by Don Hadwin). The spectral theorem says that every selfadjoint operator is unitarily equivalent to a multiplication operator. The spectral decomposition etc. fall right out of thinking about the spectral theorem from this point of view.
If you are having trouble getting started on the topic, these lectures notes from UIUC maybe very helpful: http://arxiv.org/abs/1203.2344 Spectral Theory of Partial Differential Equations by Richard Laugeson