How should I go about doing this same exact effect for an upcoming project?
In some parts it is still pictures, in some — probably video. It is better practice to make motion-design with After Effects, but you can accomplish this in Premiere.

To rotate footage in 3D in Premiere you need to add "Basic 3D" effect:

Effects > Video Effects > Perspective > Basic 3D

There you will find everything you need to rotate in x,y,x dimensions. But full video will be easier to create in After Effects. Premiere is better for video montage.

Good Luck!