The following are notes from summer schools and a couple of survey papers on coalgebra, modal logic, and category theory.
Coalgebras and Modal Logic. ESSLLI, Helsinki 2001.
An Introduction to Stone Duality. MGS 2004.
Topics in Coalgebra. Copenhagen 2004.
Coalgebras and Their Logics. Logic Colummn, 2006.
Modal Logics are Coalgebraic with Cirstea, Pattinson, Schröder, Venema. BCS, 2008.
I also have notes, from various teaching opportunities that I did not find the time to finish and polish … I put some of them here anyway, partly to motivate myself to take them up again when another opportunity arises.
Incomplete notes on Introduction to Category Theory. MGS 2017. Chapter 1 contains a long series of beginner’s exercises that should be suitable for self study if you have a mentor who can answer questions.