Note: You only get ONE chance to get MOST of these.

The Council consists of: Asari, Turians, and Salarians.
The extinct Races are: Protheans, Rachni
The Non-Council races are:Batarians, Elcor, Geth, Hanar, Keepers, Krogan, Quarians, Volus

Turians:
At the very beginning of the game, talk to people about the Spectres and Saren.

Asari:
Either talk with Captain Anderson, or with Avina[The VI about the council races].

Salarians:
Talk to Captain Anderson after the first meeting with the council.

Protheans:
CAN ONLY GET WITH TALKING WITH CAPTAIN ANDERSON!

Rachni:
Talking with Avina the VI near the Krogan Statue about the Krogans, the rebellion, and the war.

Batarians:
After acquiring the Normandy, talk with Captain Anderson. Ask him about him becoming a Specter.

Elcor
Talk to the Elcor diplomat Calyn in the embassy. The embassy is to the right of the Human Embassy.

Geth:
Ask Ashley AFTER finding her on Eden Prime.
.
Hanar
Either with the Preaching Hanar, or with the Shop Keeper that is Hanar. Ask him about his speech.

Keepers:
Ask Avina the VI near the Citadel Tower light.

Krogan:
Automatic by watching the confrontation between Wrex and the Krogan bouncer in Chora's Den.

Quarians:
Talk to Garrus in medical center.

Volus:
Talk with Avina outside the embassies.