This seems like a reasonable survey article:
[Howe - On the role of the Heisenberg group in harmonic analysis](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-3/issue-2/On-the-role-of-the-Heisenberg-group-in-harmonic-analysis/bams/1183547543.full).

Also there is a book by Deitmar and Echterhoff - "[Principles of Harmonic Analysis](https://doi.org/10.1007/978-3-319-05792-7)", which I found quite readable, but that treats coefficients in $\mathbb{R}$ only. But this book is probably more likely to be useful for graduate students.