I just joined the Adventurers League. They have told me that you can get character options (class, background, race) from the PHB and one other official D&D 5E book. I am aware of the race options, but what about class options? Is there a book that offers new class options?
Yes: For Adventurer's League players The Sword Coast Adventurer’s Guide offers new class options.
Other than the PHB, there is currently only that one official class options resources available to Adventurer's League players. The Sword Coast Adventurer’s Guide has new class options for Barbarians, Clerics, Fighters, Monks, Rogues, Paladin, Sorcerer, Warlock, & Wizard -but no new classes.
The (potentially) good news is that Xanathar's Guide To Everything is slated to have 25 new(ish) classes and/or class options. Although see what I say below on Unearthed Arcana about where most if not all of these class options are coming from. Also, the organizers of the Adventurer's League have not yet committed to using Xanathar's Guide To Everything. This is understandable, but in my opinion I find it hard to believe that the bulk of that book will not be made available to Adventurer's League players. Remember, my opinion is worth exactly what you paid for it.
As always, make sure you check the Adventurers League handouts on what specifically you can use. You can not take information from all available books buffet style. You need to pick the PHB and one extra book.
Per that Guide:
What You Need to Play
[snip] Additional options for characters are available in the fifth edition Player’s Handbook, the Sword Coast Adventurer’s Guide, the Elemental Evil Player’s Companion, and Volo’s Guide to Monsters.
The Elemental Evil Player’s Companion, and Volo’s Guide to Monsters do not have Class Options. Volo's guide seemed to focus more on Race options while the Elemental Evil Player’s Companion (pdf) is chock full of spells, and some races.
Beyond Adventurer's League Play
Since the title of your question does not specifically mention the Adventurer's League, I thought I might mention some other resources for both classes and class options. There are many more semi-official / playtest class options and at least two new classes (the Mystic and the Artificer) available via the Unearthed Arcana series of articles. The D&D Wiki has a list of non-homebrew classes, races and spells that stays pretty much up to date*.
Beyond that, there are some class options that are not official, but were made by former edition designers of D&D. Some of them are available at the DM's Guild. Or you can look at Kobold Press, or look up Monte Cook, etc.
Finally there are a whole bunch of homebrew options out there. I can't recommend this though. Not because there isn't any good stuff out there. There certainly is. But there is currently no community review of these options, and there is a whole bunch of crap to sift through before you might find any gems.
If you want to go a-hunting I recommend going through Reddit's Unearthed Arcana as they can somewhat vet homebrew stuff via Reddit's voting system. Also, it looks like DnD Beyond's homebrew section has a community review function, but they make you pay to get access to homebrew material, and that really rubs me the wrong way. (So no link to that.)
*A word of caution on the D&D Wiki.
Make sure that any resource you get off the D&D Wiki site is not labeled homebrewed. Most of the homebrew stuff from that site is just plain unworkable, not to mentioned overpowered, and not in the spirit of 5e.