DM NEEDS HELP: Learning New Proficiencies

Could someone point me to the rules for learning new proficiencies in-game? I know you have to spend 30 days and an amount of gold but I can't find the rules and I need this for my players.

EDIT: Specifically looking for skills

This is great! Thank You! But I was looking specifically for skills... Is there a rule for that or is it just considered the same as tools?

Skill proficiencies are very powerful, much more so than language or tools. I don't think there are any official rules for learning them as a downtime activity. You can gain them (assuming you're specifically talking about midcampaign) via multiclassing or through various feats, notably Skilled. Otherwise, it's up to the DM. I'd be wary of allowing it though - skills are quite powerful (PB being between a +2 and +6 depending on your level) and is one of the key differentiating features.

Here are all the downtime rules options across 2014 and 2024 rules:

2014

• Player's Handbook - Between Adventures - Training only covers languages and tool proficiencies
• Dungeon Master's Guide - Downtime Activities - This doesn't include any training but does have a section on creating downtime activities
• Xanathar's Guide to Everything - Downtime Revisited - This section on training again only covers languages and tool proficiencies

2024

• Dungeon Master's Guide - Bastions - Bastions feature a special facility called a training area. One of the options for the training area is the skills expert, but that only allows a small number of skills to be learned. These skills are also only temporary.

DMG 2024 - Marks of Prestige

Sometimes the most memorable reward for adventurers is the prestige they acquire throughout a realm. Their adventures often earn them fame and power, allies and enemies, and titles the adventurers can pass on to their descendants. This section details the most common marks of prestige that adventurers might acquire during a campaign.

The best rewards in an adventure are directly related to the circumstances of the adventure. For example, if a merchant hires the characters to retrieve a family heirloom from a long-abandoned tower, the merchant might give the deed to the tower as a reward.

[...]

Training

A character might be offered special training. This kind of training isn’t widely available and thus is highly desirable.

The character must spend 30 days with the trainer to receive a special benefit. Possible training benefits include the following:

• The character gains proficiency in a skill.
• The character gains proficiency with a tool.
• The character learns a language.

This is the only way to permanently add a Skill Proficiency to your character sheet without advancing in level.