Resources for learning Heegaard Floer homology:

Survey Articles and background:

Some nice notes on Morse Homology by Hutchings
link to ps file.

An early survey on Heegaard Floer by McDuff
link to pdf.

A few surveys on Heegaard Floer by Ozsvath and Szabo
link to arxiv.
link to arxiv.

A couple of introductions to knot Floer homology
by Manolescu.
by me and Watson.(focus on structural features of knot Floer homology)

An overview of Floer theories for 3-manifolds and their applications
by Manolescu.

Some more recent surveys of Heegaard Floer
by Juhasz.
By Hom. (focus on concordance)

Some nice introductions to Fukaya Categories
by Auroux.
by Abouzaid. (concrete focus on surfaces, where things are combinatorial)

Notes and Videos from Introductory Lecture Series:

Notes from Ozsvath and Szabo's lectures at Clay Math Institute 2004 Summer School
First set of lectures.
Second set of lectures.

Notes on bordered Floer homology from Contact and Symplectic Topology 2012 Summer School
by Lipshitz,Ozsvath,Thurston.

Notes on sutured Floer from SMF summer school on "Geometric and Quantum Topology in Dimension 3" 2014
by Lipshitz.

Lecture notes from a year long topics course I taught at MSU (notes by Matt Rathbun)
Lectures I.
Lectures II.
Lectures III.
Lectures IV.
Lectures V.
Lectures VI.
Lectures VII.
Lectures VIII.
Lectures IX.
Lectures X.
Lectures XI.

Videos of Hom's lectures on Heegaard Floer homology PCMI 2019

Videos of Levine's lectures on Heegaard Floer homology at the Winter School on Homology Theories in Low-Dimensional Topology, held at the Isaac Newton Institute, Cambridge, England, January 16-19, 2017: Part I.
Part II.
Part III.
Part IV.

Videos of Introductory Workshop for MSRI 2010 Program on Homology Theories of knots and links

Problem lists:

Kirby List.

List from Gompf's conference.

List from MPIM Conference on 4-manifolds and concordance

List from BIRS workshop "Synchronizing Smooth and Topological 4-Manifolds"

List from BIRS workshop "Thirty Years of Floer Theory for 3-manifolds"

List from Nearly Carbon Neutral Geometric Topology Conference 2020