I've just installed Drupal and I'm familiarizing myself with all of the core features. The first thing that's not obvious to me is When I add a block in the structure screen I don't have any ability to choose the block type, I only have an area in which to add HTML. How do I create more useful, pre-built blocks of content?
You can create blocks in many ways.
The default way you know already
Views module allows you to create blocks with lists of content, or practically anything you might want
Nodeblock module allows you to use nodes as a blocks if you wish to do so
Boxes are a vision how the built-in blocks should have been implemented in the first place
Many modules ads their own blocks, but it depends on functionality you need on your page; if you will enable a module that provides it's blocks, you will see these blocks on the structure page just like you see the ones that are already there
And you can always write your own module that will add some blocks, see Examples for Developers to know how to do it
Adding a block is not meant for adding any specific type of block. It is used to add a custom content with in a block region.
Blocks are created by modules. For example the core user module creates the User Login Block.
Another example would be views. When you create a new view of the type block it would be displayed with the block page and you can place the same in any of the regions.
When you add a new module and if the module has defined a block then it would appear with in the structures->blocks page.
You can also define a block with in your custom module.