The game offers a few methods by which a character can gain proficiencies, aside from those chosen at first level for class and background, and any bonus proficiencies learned due to specialisation in a class.
The Skilled feat (PHB pg.170) can be chosen by characters using the optional feat rules, and simply grants the character any combination of 3 additional tool or skill proficiencies, chosen freely.
The Weapon Master feat grants the character proficiency with 4 additional simple or martial weapons (alongside a +1 to strength or dexterity) - though the option is honestly a bit weak compared to just multiclassing (see below) one level in a martial class, unless you desperately want a specific weapon proficiency and you can't afford to lose a level from your main class progression.
The feats Lightly Armored, Moderately Armored and Heavily Armored respectively grant proficiency with Light, Medium and Heavy armour.
If a character multiclasses into a new class, they may pick up some new proficiencies, which might include a new skill proficiency.
When you gain your first level in a class other than your initial class, you gain only some of new class's starting proficiencies, as shown in the Multiclassing Proficiencies table.
Specifically, multiclassing into bard, ranger, or rogue allows the character to choose an extra skill proficiency from the relevant class skill list. Multiclass bards additionally learn proficiency with a musical instrument, and multiclass rogues also learn tool proficiency with Thieves' Tools.
All multiclasses bar sorcerer and wizard otherwise offer some proficiencies with weapons and armour, though notably no multiclass choice grants heavy armour proficiency by default (though if choosing cleric, several of the domains grant bonus proficiency with heavy armour as a class feature, which would still apply when multiclassing).
Chapter 7 of the Dungeon Master's Guide is all about treasure and reward, and the Other Rewards section of it guides the DM in giving characters rewards for their exploits beyond simple money and magical items. Specifically, page 231 describes the offer of special training as a reward for a character, which might grant them proficiency in a skill:
A character might be offered special training in lieu of a financial reward. [...]
[...] Possible training benefits include the following:
- The character gains proficiency in a skill.
So a DM might offer characters the chance to learn a new skill as a reward, in addition to or instead of ordinary treasure.
If a character actually only wishes to learn a new language or tool proficiency, rather than the more valuable skill proficiency, this is given as an option when using downtime for training:
You can spend time between adventures learning a new language or training with a set of tools. Your DM might allow additional training options.
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.
This version of the downtime training rules requires an extremely long period of total downtime at 250 days, which probably makes it not a realistic option in many campaigns; however, Xanathar's Guide to Everything offers a reworked version in its Downtime Revisited section on page 134, which requires considerably less time:
Receiving training in a language or tool typically takes at least ten workweeks, but this time is reduced by a number of workweeks equal to the character’s Intelligence modifier (an Intelligence penalty doesn’t increase the time needed). Training costs 25 gp per workweek.
A workweek in this context is 5 days of effort, so generally about 5 times faster than the PHB version (though the cost is similar).
Honourable mention: 2nd level bards
At 2nd level, bards gain the Jack of all Trades class feature:
Starting at 2nd level, you can add half your proficiency bonus, rounded down, to any ability check you make that doesn’t already include your proficiency bonus.
This isn't full proficiency in any skills, but it does make the bard somewhat better at using any skills they aren't already proficient in, and can be had for a 2-level drop in bard. (3 levels grants Expertise in a couple of skills and 2nd level spells, which might be an attractive option for a character who has already taken 2 bard levels.)