There aren't too many published materials that contain official classes (classes here meaning sub-classes and archetypes included) but they are scattered resources.
What are the official, 1st-party, WotC-published classes and which resources are they found in?
For the purposes of this question "unofficial", playtest material such as Unearthed Arcana is excluded and will be asked as a separate question.
- What are the playable D&D races in 5e?
- Where do I find the "official" rules for D&D 5e?
- Is it necessary to purchase all the D&D 5th books to have access to all character classes etc.? (even though the answer to this linked question just happens to more-or-less answer this question, they are different questions)
and meta discussion and approval for this question here.