Entity API and Contrib Modules When using the Entity API, is it fairly simple to interact with Taxonomy, Path/Pathauto, XML Sitemap, and Views? If so, what parts of the API should I be looking at?
In response to the original question and to electblake's answer, I would second the fact that the Entity API contributed module is separate and distinct from the core entity system. It is meant to augment the core system by providing a more robust entity controller (the piece of code used to create, load, save, and delete entities), provide some automatic token generation, and allow modules to define metadata for their entities' properties.

electblake mentioned that the core entity system includes a class for querying entities in the database based on their property and field data called EntityFieldQuery. This is in fact the recommended way for making entity queries. The Entity API isn't actually meant to be a querying tool. However, it provides in my opinion the other missing piece for building robust contributed modules using entities and fields - the entity metadata wrapper.

As I mentioned, the Entity API module provides a hook that lets you define metadata for your entity's properties (i.e. $node->uid, $node->created). It combines this information with what it knows about field definitions and instances. It then lets you "wrap" an entity to make it easier to access and manipulate field data and (what I'll call) referenced entity data through the wrapper object. One of the main things this has made easier for me is not having to worry about the language of entities when I'm trying to access their field data. Without the wrapper, you need to know the schema array of the data you're trying to access or manipulate. Using it, you can simply access field data like any other property.

Additionally, it makes it easy to access the data of referenced entities. In Drupal Commerce, I make extensive use of this functionality since a couple of our entities serve primarily as containers of references to other entities. An example approaching the usefulness you mention in your original post would be to have quick access to all the taxonomy terms referenced by a node. A simple way to do this with the wrapper would be to wrap the node and then loop over your taxonomy reference field's data to access the fully loaded term data. It's hard to demonstrate this without including extensive code snippets, but you can easily see how I put this to work by grepping the Drupal Commerce source for entity_metadata_wrapper.

The jury's still out on how useful or widespread the Entity API will be, but as it's a dependency of Rules 2, I doubt it's going anywhere soon. This may be a little too much information based on your response to electblake, but my hunch is the information should be useful for additional people who find this question when searching for more information on the Entity API. : )