How should I go about doing this same exact effect for an upcoming project? A client has tasked me to create a How-to-video for Hotel/Flight booking app. They sent me this video and told me to that they would like the same video. The problem is I have no idea what kind of effect/plugins were used to achieving the effect.

Was it an actual video recording of the booking website or were those just image graphics? I'm primarily interested in the 3D effect. Is it possible to mimic the effect using Adobe Premiere alone or do I need a post production program such as After Effects? I would greatly appreciate helpful inputs.
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!