How can I delete or shift certain sections of the Timeline? I'm using a single Blender file to create a 2D slideshow (with animated graphs and stuff). So all the "slides" are in this file. They are of course not actual slides, but sections of the Timeline (so the entire slideshow is run by pressing alt+A). How can I make space in the Timeline to "insert" a new "slide", or how can I "delete" a "slide"?
In the Dope Sheet editor, you can select a specific keyframe, or several of them (use B to border select), and move them with G. You can delete the keys with Suppr or X.

To go further, maybe you could create an Action for each slide, that's a great way to manipulate them using the NLA editor (basically, each slide will act like a movie clip in a video editor). NLA Editor Action Editor