Is skill retraining explained in any of the 5th edition D&D core rulebooks?
From the reading that I've done, there is no explicit explanation of a "retraining" mechanism in 5th edition D&D.
Other Courses of Action
- The Skilled Feat
A way to avoid undoing skill choices is to make sure that you can train more skills after Level 1. Conveniently, there is a feat called Skilled that gives players the options to train themselves in three more skills than they are trained in through their class and race. Considering how often skills are utilized for the group's benefit in D&D, this is a very viable feat to have a character take. The feat is described as follows (PHB, pg. 170):
You gain proficiency in any combination of three skills or tools of your choice.
- Learning from a Willing Instructor
The Player Handbook describes the manner in which a player's character may learn a new language or tool, a concept which could reasonably be extended, if the DM sees it appropriate, to include new skills as well (PHB pg. 187):
First, you must find an instructor willing to teach you. The DM determines how long it takes, and whether one or more ability checks are required. The training lasts for 250 days and costs 1 gp per day. After you spend the requisite amount of time and money, you learn the new language or gain proficiency with the new tool.
As I said, the PHB never specifically mentions that learning new skills can be done as a down-time activity. You could reasonably apply the above rule to allow players the option of learning skills from one another. Judging from the wording of the Skilled feat, described earlier, the PHB indicates that proficiency with a certain set of tools, which are specifically allowed to be taught and trained through a willing instructor according to this excerpt, and proficiency in a certain skill are valued at roughly the same level.
- Multi-classing for Additional Skills
Under the "Proficiencies" sub-header in the "Backgrounds" section, the PHB sets forth a useful rule to keep in mind on "double-dipping" proficiencies / skills (PHB pg. 125):
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
"A different proficiency of the same kind" seems to indicate that the character is allowed to train in any of the list of proficiencies provided on the player's character sheet as a substitute for their "double-dipped" proficiency. For all of the below multi-classing options, it is important to remember that if you multi-class into a class that offers many of the same skills as your original class, you are allowed to substitute those "double-dipped" skills you are forced to take with any other skill on your character sheet.
Short story shorter, if you want to train in as many skills as possible, then it is best to make sure that your character's original skill choices overlap heavily with the skill choices offered by the class you are multi-classing into (particularly, Rogue or Ranger) so that you have the whole list of skills available to train your character in, instead of just that class's predetermined set of skills.
Training your character as a 1st-level Bard allows your character to train into three new skills. Further, selecting the College of Lore archetype allows your character to train into three additional skills.
Training your character as a 1st-level Rogue allows your character to train into four new skills, selected from a predetermined set of skills.
Training your character as a 1st-level Ranger allows your character to train into three new skills, selected from a predetermined set of skills.