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I skimmed the PHB, but I didn't find any information on "retraining" or any ways to "undo" character advancement. Is retraining (especially of classes and class features) available in any of the core books? How does it work?
I'm curious about the possibility of retraining a standard Paladin into a full-on Oathbreaker.
The only options for training of any sort I see at all are:
The PHB offers rules for multiclassing (p163) but this isn't so much "retraining" as it is training in a different field entirely.
The PHB (p187) presents training in a tool or language as a possible downtime activity. It says, "Your DM might allow additional training options." It takes about as long as a public school year in the US to complete one training option.
The DMG (p131) presents optional rules for training to gain levels. That is to say, with these optional rules you must spend gold when you reach the required XP value to gain a level. This training lasts about 2 days per level.
The DMG (p231) also presents rules for granting training as an "alternative reward." That means this option is presented as an alternative to magic items, so you're not going to be able to just go out and do it. You can spend the same amount of downtime that the PHB requires for training to learn a new skill, gain a feat, or gain inspiration daily for about a week. You'd have to ask someone else how 7-10 days of inspiration compares to a permanent skill or feat, let alone would be worth the 9 months of investment to get it.
However, keep in mind this is 5e, not 3e, and the game takes great pains to remind readers that the DM can use whatever options they want in their game, including those they devise themself or adopt from other games.
In the core rules and SRD, no, there's no way to undo training, so choose carefully. Once you've made a class selection, you're keeping it unless there's some serious reincarnation shenanigans involved (or you're a druid/cleric/paladin and lose your powers by offending your god/philosophy).
In D&D Adventurer's League, you're permitted to rebuild your character up to the end of Level 4 (Player's Guide, page 4, "Character Rebuilding"), though your name must stay the same and any faction renown is lost if you change factions.
In the Character Options chapter of Tasha's Cauldron, we see:
Each character class involves the choice of a subclass at 1st, 2nd, or 3rd level. A subclass represents an area of specialization and offers different class features as you level up. With your DM’s approval, you can change your subclass when you would normally gain a new subclass feature. If you decide to make this change, choose another subclass that belongs to your class and replace all your old subclass features with the features of the new subclass that are for your new level and lower.
Tasha's goes on to give guidance for how a DM might handle such a change in-universe, and the details are the subject of this Q&A: How exactly do the rules on changing your subclass from "Tasha's Cauldron of Everything" work?
Specific to your given change, standard paladin to Oathbreaker, one of the examples for a sudden change in subclass given in Tasha's is almost exactly what you're looking for:
An Oath of Devotion paladin failed to stop a demonic horde from ravaging her homeland. After spending a night in sorrowful prayer, she rises the next morning with the features of the Oath of Vengeance, ready to hunt down the horde.
