There are two different ways a race could be "official": it could be "official" in the sense of fully-tested and legal for sanctioned organized play campaigns, and it could be "official" in the sense of being published in 1st-party material from Wizards of the Coast. Which one is relevant depends on why you're looking for an "official" race.
BR: Basic Rules, Chapter 2: Races
SRD: System Reference Document, 5e, "Races" at p. 3-7
PHB: Player's Handbook, Chapter 2: Races
DMG: Dungeon Master's Guide, "Creating New Character Options" at p. 285-287
AI: Acquisitions Incorporated, "New Race: Verdan" at p. 72-74
EE: Elemental Evil Player's Companion, Chapter 1: Races
GGR: Guildmaster's Guide to Ravnica, p. 12-21
MToF: Mordenkainen's Tome of Foes, throughout Chapters 1-5
MOoT: Mythic Odesseys of Theros, Chapter 1 > Races
LR: Locathah Rising, an Extra Life fundraiser from WotC
OGA: One Grung Above, an Extra Life fundraiser from WotC
PotA: Princes of the Apocalypse, Appendix A: Genasi
PS:ZEN: Plane Shift: Zendikar, Races of Zendikar at p. 7
PS:INN: Plane Shift: Innistrad, Life on Innistrad at p. 7
PS:KLD: Plane Shift: Kaladesh, Races of Kaladesh at p. 15
PS:AKH: Plane Shift Amonkhet, Races of Amonkhet at p. 12
Note that Plane Shift occupies a strange shade-of-grey space: it's "compatible with" D&D, published by WotC, but written by the Magic: the Gathering team instead of the D&D team (though the author of all of them was a former D&D team member). Notably, it comes with a disclaimer substantially similar to the one provided with UA articles:
The game mechanics in this supplement are usable in your D&D campaign but are not fully tempered by playtests and design iterations. For these reasons, material in this supplement is not legal in D&D Organized Play events.
SCAG: Sword Coast Adventurer's Guide, Chapter 3: Races of the Realms
TP: The Tortle Package, an Extra Life fundraiser from WotC
UA: Unearthed Arcana articles, link and name specific article when listing
UA articles are presented on WotC's website with the following notice:
You can think of the material presented in this series as similar to the first wave of the fifth edition playtest. These game mechanics are in draft form, usable in your campaign but not fully tempered by playtests and design iterations. They are not officially part of the game. For these reasons, material in this column is not legal in D&D Organized Play events.
VOLO: Volo's Guide to Monsters, Chapter 2: Character Races
WGtE: Wayfinder's Guide to Eberron, Chapter 3: Races of Eberron
E:RftLW: Eberron: Rising from the Last War, Chapter 1: Character Creation
Note that Wayfinder's Guide to Eberron and Eberron: Rising from the Last War now redundantly feature most of the same new races, though the latter reprints the goblin, bugbear, and hobgoblin races, and includes a new version of the orc race. WGtE was originally published as a "living document", and was later updated with the versions of the included races as they appear in E:RftLW.
EGtW: Explorer's Guide to Wildemount, Chapter 4: Character Options, Races
Best Answer
Sources where they first became available to players:
PHB = Player's Handbook
MM = Monsters Manual
MToF = Mordenkainen's Tome of Foes
SCAG = Swords Coast Adventurer's Guide
VGtM = Volo's Guide to Monsters
GGtR = Guildmasters' Guide to Ravnica
WGtE = Wayfinder's Guide to Eberron
DMG = Dungeon Master's Guide
EGtW = Explorer's Guide to Wildemount
Standard Languages:
Standard Languages (Ravnica):
Exotic Languages:
Race/Class-Specific Languages:
Forgotten Realms Human Languages (requires GM permission):
(All in SCAG)
Eberron's Common Languages
Monstrous Languages (available via Favored Enemy or Training):
Book of Vile Darkness
Wildemount's common languages