Where to start?

Schanuel's conjecture, the existential closedness problem and the Zilber-Pink conjecture are open problems that have seen a lot input coming from different areas of mathematics (especially in recent years), and so it can be difficult to figure out how to begin learning about them. There are many resources available, but they can be hard to find if one doesn't know where to look. So here I want to point out some of them to help anyone who is interested.

I have also compiled a list of relevant articles.

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 knowledgeable

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 (in particular the ones by Jonathan Pila and Tom Scanlon).
Three conjectures in number theory
Survey article giving an overview of the problems.

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.
I think this is a great place to start if you are a PhD student. It helped me a lot to get a an idea of the landscape, and what were the main objects everyone was talking about.
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.