Reference on Fourier analysis on compact groups

Chapter 5 of Folland's A Course in Abstract Harmonic Analysis should have what you need -- it is quite a short treatment, but it seems to be complete, provided that one is happy to fill in (routine) details in a narrative rather than go for the style of "Lemma 2.1.2, Lemma 2.1.3, Definition 2.1.4, Proposition 2.1.5, Lemma 2.2.1, etc".

I think there is also a readable treatment in Deitmar's A First Course in Harmonic Analysis, although the approach in that book is tailored to an audience without Lebesgue integration and hence might have some proofs which are non-geodesic work-arounds.

For a reference which 1) has what you need, 2) is short and elementary (only undergrad point set topology needed, e.g., Ascoli/Arzela Theorem) yet includes detailed proofs, 3) is very clear and pedagogical, 4) is free; I think you will have a hard time finding better than the note "Haar Measure" by Joel Feldman, on his UBC webpage.