Customizing Photo Galleries For More Page Views and Organization

I currently use the standard WordPress galleries, and they work just fine for me. Recently I was wondering if it was possible to customize them for more page views and organization.

What I would like to happen is that I can insert a photo gallery into a post and then when you click on an image thumbnail you are taken to a new page to view the larger image. When you click on the larger image you are then taken to the next image in that gallery, and so on.

Here is an example of what I mean, but I don’t think it’s on WordPress: https://www.actionfigureinsider.com/gallery2/album/E3-2013-Accessories/1371193736-1.php

Then for the organization part of this I would like to do something like this: https://www.actionfigureinsider.com/gallery2/

With this I could then organize photo galleries by event, if they were associated with a particular event or trade show.

If the organizational part is not possible, that’s fine. But what I would really like to do is make the multiple page views for each photo on my posts (the first link).

If someone could point me in the right direction to get started that would be great. I have already looked into the NextGen gallery plugin, but it appears that it’s not going to work for me, as the images are not indexed by Google. I would really prefer to do this with the default WordPress gallery, but am open to all options.

Thanks in advance.