Best slider/gallery plugin for my needs?

I need a plugin recommendation. I would like a gallery plugin with a slider to mimic the following design I mocked up in photoshop: Example image

Basically, I need a plugin to allow me to display my portfolio. I need four categories: character design, illustration, book covers, and personal. When the user clicks on one of these categories, I need a double row of images to pop up to the right, and the first image in the group to appear with navigation arrows. I am not sure if a need a gallery plugin or a slider plugin to do this.

Additionally, it would be great if I could get a responsive design, something that people can scroll through on their tablets or iphones.

I usually hand-code everything, so if something doesn’t fit my needs, is there a way to hack into an existing plugin to get what I need done?