District 13, as a general whole, is motivated by a combination of survival and resentment against the Capital.
It is safe to assume that there is some amount of sympathy for those subject to the Capital's cruel and oppressive treatment, as well, but there is little direct evidence of that in the books. The fact that District 13 was part of the original rebellion against the Capital gives some pretty good indirect evidence to support that, though.
After the rebellion, when District 13 was able to isolate themselves, the isolation coupled with outbreak of disease pushed them to the edge of extinction. While they recovered from the threat and managed to re-establish their safety, their gene pool was dangerously thinned.
They also felt confined, forced to live almost exclusively in underground bunkers, with only short periods allocated for surface exercise.
These are the primary factors for the motivation of the District as a whole. They wanted freedom from having to hide underground. They wanted revenge on the Capital for putting them there, and for creating the conditions that led them to rebel in the first place. They want an influx of new people to stabilize and reverse their population decline. They also (probably) want to help the other victims of the Capital.
President Coin has her own personal agenda. She wants power, and demonstrated a willingness to sacrifice whoever was in her way. This made her not significantly different than President Snow, both in motivation and in methodology. However, I do not believe it fair to assume that her motivation is reflective of the whole District.