To show that $E\to M$ is a $G$-bundle (say in the fin-dim manifold category), one needs to check local triviality using some charts on $M$ and $E$ and that transitions between them are controlled by the group $G$. However, if you have a principal $G$-bundle on your hands, you can easily use the associated bundle construction to build the total space and the projection maps that satisfy the bundle property given only how $G$ acts on an abstract typical fiber.