Premiere Pro - How to make a slideshow/compilation
Premiere Pro has a number of default transitions under the "video transitions" tab in the effects pane (usually located in the bottom left of the window). Are those what you are looking for? If you have experience with After Effects, you could also use AE's more robust effects library to animate interesting effects for your videos.

Based on your included image, it looks like you might be interested in wipe transitions.