Mass Effect: Andromeda only includes five of the races we know and love from the franchise (humans, turians, asari, krogan and salarians) and only introduces two new species native to the Andromeda galaxy (the angarans and the kett — though your enemies, the kett, may or may not count for [SPOILER] reasons). So what about the other races from the Milky Way?
You’ll never encounter the other races during the game — or even hear anyone talk about them. It’s not until the scene that follows the end of "Meridian: The Way Home" that you can find your first clue.
On the Nexus, while you’re running around talking to your crew, you’ll bump into Rynn Gee, who has a message she wants you to hear. It seems the ark that was carrying the quarians — and the elcor, drell, volus and hanar — is still out there somewhere. And they’re broadcasting a signal warning the Initiative to keep away.
So, the short answer is that the other Milky Way species don’t appear in the game. But the longer answer is that they are mentioned in the (effectively) after-credits stinger sequence as a teaser for what’s to come.