I'm looking for a good starting book on the subject which only assumes standard undergraduate background.
In particular, I need to gain some confidence working with properties of Haar measures, so I can better understand the spaces $L^{p}(G)$ for a locally compact group $G$.
For some perspective on what I currently know, I can't yet solve this problem: Verifying Convolution Identities
Something with plenty of exercises would be ideal.
I suggest the short book by Robert, "Introduction to the Representation Theory of Compact and Locally Compact Groups" which is leisurely and has plenty of exercises. The only prerequisite of this book is some familiarity of finite dimensional representations.
A second book you should look at is Folland's "A Course in Abstract Harmonic Analysis", which is more advanced, and requires more experience with analysis (having seen Banach spaces is not a bad thing), but the advantage of this book is that it has very clearly written proofs, that are easily to follow (I do algebra mostly, and I find many analysis tracts a bit opaque in this regard). Unfortunately, this book does not have exercises, and should be approached once you have plenty of examples in mind.
Donald Cohn's "measure theory" has a large number of exercises on the basics of topological groups and Haar measure, but it doesn't do representation theory or much else on locally compact groups except an introduction.
For a historical view, see
Mackey, George W. Harmonic analysis as the exploitation of symmetry—a historical survey. Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 1, part 1, 543–698.
Dym and McKean's book Fourier Series and Integrals is very well-written, begins with standard undergraduate harmonic analysis on $\mathbb{S}^1$ and $\mathbb{R}$, and devotes the latter half to Fourier analysis on finite groups and some good examples of compact groups ($SO(3)$ in particular).
Since I have not yet read it (it was recommended to me by my advisor, so I am in the process of acquisition), I am not sure at what level Helgason's book Topics in Harmonic Analysis on Homogeneous Spaces is written. You might consider checking it out to see if you can follow his arguments, or at least understand statements of results.
