Starting Languages: Your character knows Common and one other starting language, plus a bonus language for each point of Intelligence, Wisdom, or Charisma bonus.
Number of languages spoken is much more a matter of experience and (generally childhood) exposure than mental ability. Unfortunately, as I hinted at earlier, D&D just doesn't handle this aspect well, so you might be stuck with it. Maybe you could tie this in to backgrounds or custom culture or something.
There would be three main language groups:
Corrian (language of the dwarven empire)
Westongue (language of the western cultures)
Greenspeech (language of the eastern cultures)
Again the hierarchy is language group (actually family) > language > dialect. You don't have dialects as direct members of a language group. Those three above should be languages, full stop.
Each language group has a few dialects. When a character is first interacting with someone who speaks a different dialect, they may make a DC 15 Wisdom (Insight) check. If they succeed, they may communicate as if speaking Common (see below). If they succeed at a DC 20 Wisdom (Insight) check, they may communicate as if proficient in the same language.
No, no. (Neighboring) dialects are generally mutually intelligible. Also don't make ability checks required for potentially common situations, that's why people generally bin this whole topic and use a common tongue or universal translator in games. At most, people speaking different close dialects should have no trouble communicating information, but they recognize the other's dialect (without a Deception check of course)—which may color NPC attitudes for Persuasion and other such checks.
If you want to get into dialect continuums, you can give a threshold past which dialects in the continuum impose disadvantage on communication of information, and past which dialects are no longer mutually intelligible, with a small (like, -2 tops) penalty to language checks for dialects that are 2–3 steps apart. Again, a distance of 1 should not be a problem. (Defining dialect distances is up to the setting designer!)
Common
Common is a pidgin language made up of bits of all spoken dialects.
Incomplete Vocabulary: When speaking Common, a character suffers disadvantage on all Charisma checks that require speech.
This should probably be topical. Common would be a great language for general commerce, travel, and such, with no penalty. But Common would not be a good language for nuanced diplomatic negotiation (which opens another sub-area of
prestige languages and the influence that might have).
@Davies made good suggestions.
Other Languages
Celestial
Spoken by celestial outsiders.
Language of Healing: When a character proficient in Celestial casts a spell that heals hit points, they may add 1 to the total amount of hit points healed.
Minor but very colorful. A good combo for language bennies.
Fiendish
Spoken by devils and demons.
Language of Lies: A character proficient in Fiendish may use Intelligence instead of Charisma when making a Charisma (Deception) check.
Slight potential for min-maxy abuse, but also very colorful.
Sylvan
Spoken by the fey.
Language of Magic: A character proficient in Sylvan gains advantage on Intelligence (Arcana) checks made to identify spells.
I thought Draconic was the language of magic? Fey is all about enchantements and illusions.
Dovorian
The dovorian language is spoken by a large empire of orcs to the south of the region.
War Speech: A character speaking Dovorian may make a Charisma (Intimidate) check even against those they do not share a language with.
Very colorful.
Kekoko
The kekoko language is spoken by a wide array of lizardfolk, tortles, and other reptillians who live in the vast swamps north of the region.
Like a Drum: The kekoko language contains words that can be spoken incredibly loudly; those proficient in Kekoko may use the language to communicate at double the range spoken words would normally travel.
Ability to speak loudly—assuming a language falls into the usual parameters of pulmonic production—is more dependent on physiology than the speech sounds of a language. But, have you heard of
whistle speech?
Also, most reptiles & lizards don't have particularly loud vocalizations. Mammals & avians (especially
parrots and the
white bellbird), on the other hand....