Izakaya/ First Nation Food?
For Izakaya, we loved Hapa Izakaya on Robson—noisy, frenetic, fun and delicious. As far as Frist Nations food, the only place I know of was Liliget, which has closed as Bob Mac mentioned. Many local restaurants, though, feature some First Nations-inspired dishes: alder-planked salmon, bannock, etc.
By "First Nation" or "Nations" food I am assuming reference is being made to local coastal "Aboriginal" or "Indian" food.
I expect the emphasis would be regional cuisine cooked in a fashion as least similar to how the original inhabitants did. Of course that might be interpeted in a very liberal fashion.
Smoked and barbequed salmon, etc. etc.
There used to be one place called the Liliget Feast House which purportedly specialized in this but a quick google search informed me that it is no more.
I am not sure if the Rain Tree is still in Vancouver or not. I remember going there years and years ago. It was not much with pretty bad service.