I recently did a deep dive into the topic, so here's a list of things I came across that helped me:

[1] Teleman 2005: https://math.berkeley.edu/~teleman/math/RepThry.pdf

[2] Khonvanov List of Repr Theory Resources http://www.math.columbia.edu/~khovanov/resources/

[3] Huang 2010, "Fourier-Theorietic Probabalistic Inference over Permutations"

[4] Woit, Topics in Representation Theory Course, http://www.math.columbia.edu/~woit/repthy.html

[5] Gallier 2013, "Spherical Harmonics and Linear Representations of Lie Groups" https://www.cis.upenn.edu/~cis610/sharmonics.pdf

Not a standard source for algebraists, but Percy Diaconis's book, mentioned in other comments, is an excellent introduction. Also contains applications in probability.