Where to start?

For the mathematically inclined

Resources with a focus on motivation rather than detail. Easy to read. 

• Check out Vahagn Aslanyan's blog. 

• What is Existential Closedness?
  A description of the existential closedness problem for exponentiation intended for general maths enthusiasts. 

For the mathematically proficient

Survey works describing the three problems, and including relevant references to current methods and results. 

• Lecture notes for the Arizona Winter School 2023 on Unlikely Intersections, written by Laura De Marco, Jonathan Pila, Thomas Scanlon, and Jacob Tsimerman. 

• Three conjectures in number theory
  Detailed survey article describing the problems and the partial results for the case of exponentiation. 

• Unlikely Intersections in Shimura Varieties and Beyond: A Survey
  written by Christopher Daw.

Books

Aimed at graduate students and researchers.

• Some Problems of Unlikely Intersections in Arithmetic and Geometry
  Umberto Zannier with appendices by David Masser, Annals of Mathematics Studies 181, Princeton University Press, Princeton, 2012.  

• O-minimality and Diophantine Geometry
  Gareth Jones and Alex Wilkie (eds), LMS Lecture Notes Series 421, Cambridge University Press, Cambridge, 2015.

• Around the Zilber—Pink Conjecture
  Philipp Habegger, Gaël Rémond, Thomas Scanlon, Emmanuel Ullmo and Andrei Yafaev (eds), Course Notes CIRM, Panorama et Synthèses 52, Société Mathématique de France, Paris, 2017.

• Point Counting and the Zilber—Pink Conjecture
  Jonathan Pila, Cambridge Tracts in Mathematics 228, Cambridge University Press, 2022.