Ok I'll answer this from a practioner's perspective rather than a purist's (which I am solidly not).
Below are the books that influence my understanding of this space, listed in chronological order of when I bought them.
TL;DR - buy Hull and Wilmott's books & read in that order. But also search for papers by Duffy after having read Hull.
- Options, Futures, and Other Derivatives, John Hull
I'd consider this essential reading for anyone interested in derivatives. It covers tonnes of ground you need in general but specific to PDE pricing it has: Ito's Lemma in sufficient detail for you to be dangerous, binomial trees in sufficient detail for you to get the idea but also see their limitations and, finally, finite difference methods / illustrations for pricing European and American options in both explicit and implicit FDM. Hugely useful for the junior practitioner - but make sure to implement explicit & implicit FDM in code yourself before you move on! You won't know it otherwise.
- Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach, Daniel Duffy
This book is for the serious purist - method of lines, operator splitting, etc. Powerful techniques heavily influenced by some of the greats of early PDE study (lots of Soviets) but as a practitioner text I found it too dry / abstract to be of strong utility in my day job. Duffy is a very good author (and a very nice guy, having conversed with him over the years) with a number of papers I like (specific to PDEs you need to read The Alternating Direction Explicit (ADE) Method for One-Factor Problems - this is my go-to method for general PDE option pricing).
- Paul Wilmott on Quantitative Finance 2nd Edition: 3 Volume Set
This is where you can really explore PDE option pricing as he extends into path-dependent options, auxiliary variables, dimension reduction techniques etc. Much like Hull, this book is direct and accessible with sufficient mathematical precision to avoid ambiguity. Huge practical value and a lovely book to read. Lots of exciting ideas to try out.
I hope this list is helpful for you.