Together with Marius Crainic and Rui Loja Fernades, we wrote an introductory book to Poisson Geometry. The book grew out of several sets of lecture notes that we prepared for master- and graduate-level courses on Poisson Geometry, which we taught at our home institutions, and for minicourses at various Poisson geometry schools.
You can download the book from here, or you can order a physical copy from the AMS website.
I taught Poisson Geometry, as part of the Dutch MasterMath. The recordings from 2021 are available here (password: vB6a), and from 2023 here (password: Pg23).
For a shorter introduction to Poisson Geometry, watch the minicourse I gave at the Summer School on Poisson Geometry 2018, at the Fields Institute.