What is the Republic?
The Republic is your environment and the people around you. It is your family, your neighborhood, your city, your state, your country, your nation, your race, your homeworld. You train for the Republic because otherwise you are training for yourself which is a selfish attachment to the ego. When connected to the Force you lose your ego and you realize that you are no different than those around you. This develops empathy, allowing you to feel what others feel. And if you could feel the pain inflicted on them, either physically, emotionally, or verbally, you would want to stop it just like you would want to stop any pain attacking you.
Once you reach that point, skin color, nationalities, borders, races, and any other tags we give are simply a way to further divide us instead of bringing us all together.