Easier reference for material like Diaconis's "Group representations in probability and statistics"

I have a chapter on this in my book Steinberg - Representation theory of finite groups. Sorry for the self promotion. It is intended for advanced undergrads. I basically focus on the abelian case, giving the upper bound lemma on convergence rates and the description of the eigenvalues for this case only. I do one explicit computation (I don't have the book in front of me right now, but I think I do it for the lazy random walk on the hypercube). Also, if memory serves there is an exercise on what to do if the probability measure is constant on conjugacy classes for non abelian groups,, but maybe that was in the section on eigenvalues of Cayley graphs.)

An alternative is Harmonic Analysis on Finite Groups Representation Theory, Gelfand Pairs and Markov Chain by Tullio Ceccherini-Silberstein, Fabio Scarabotti, andFilippo Tolli, but I find it no easier than Diaconis except that it may expect more of an algebra background than a probability background.

Recently, Benjamin Steinberg write "Representation Theory of Finite Groups: An Introductory Approach". Is a modern and very nice written course. It include Fourier analysis on finite groups and probability and random walks on groups.

I would also recommend "Introduction to Representation Theory" by Pavel Etingof et. al. It come from lectures notes that Etingof writed for MIT undergraduate students. It contain a bunch of subjects that students can take for a project (mostly abstract subjects, like Schur-Weyl duality, quiver representations, category theory, homological algebra).

"Representations and character of groups", by Gordon James and Martin Liebeck is another nice reference for an undergraduate course. The last chapter contain applications to molecular vibration.

This 74-page paper in Journal of Machine Learning Research (by Huang, Guestrin, and Guibas) --

http://www.jmlr.org/papers/volume10/huang09a/huang09a.pdf

-- is an amazingly useful and undergrad-friendly intro to the representation theory of the symmetric group (in a very surprising venue). Also, the fact that its applications are to machine learning and statistics looks very appropriate for OP's question.