It's been quite some time since I thought about harmonic analysis books beyond the few favourites that I've found, but I seem to recall getting the impression that harmonic analysis in the sense of Stein is an entirely different beast to Deitmar (Stein is really analysis in the classical sense, focusing on harmonic functions of a complex variable, while Deitmar is representation theoretic). Perhaps I'm misremembering, but I have the distinct impression that undergraduate me got pretty confused by this!
Based on you wanting to understand the representation theoretic version, Deitmar is a good call. Assuming you mean his Principles of harmonic analysis, that emerged as my favourite when I was learning it. I also liked Folland's book (Introduction to harmonic analysis, or something similar) and Terry Tao's online notes on the Peter--Weyl theorem. Both should be easily googleable.
I don't really think spectral theory is a prerequisite. The treatments of the compact case which I know use the spectral theorem in an essential way, but you really don't need much spectral theory for this: it's just a simple generalization of a familiar fact from finite-dimensional linear algebra. Much more important is a solid understanding of the representation theory of finite groups, as this is what the compact case is analogous to. Measure theory would also be useful, but I don't really think you need a proper course in that either; more just familiarity with the Lebesgue measure, as this is the measure generalized by Haar measure.
The abelian case is kind of funny in that most treatments are really more analogous to classical Fourier analysis, with the "representation theory" side of things being fairly trivial as everything is one-dimensional. I haven't thought about this case at all since I learned it, so I won't say too much.
On the other hand, the non-compact non-abelian case is extremely difficult. Honestly, I wouldn't bother trying to learn anything going into the proofs at all -- just the statement of the Plancherel theorem should be fine. The only proof that I've seen the proof of this written down is in Dixmier's books on $C^*$ and von Neumann algebras, both of which are extremely difficult books, and it requires pretty much all of the both of them. As long as you're happy to learn statements, Deitmar's book has a nice enough chapter on this.