JohnnyTs response was really good. I'll just add some ideas from my own experience.
If you are looking for practical projects to work with, a good place to start is to try to create a document management system where you integrate with the Office applications and use templates, different content types and metadata columns and/or managed metadata. This can all be done through the web interface, and you will end up getting a broad overview over many aspects of SharePoint. Make sure you also hook up the Document Information Panel in the Office applications (this is where you set document metadata when working from Office) - there are a lot of pitfalls you can end up stepping in, but when you've done it once you will have good control on the administrative side of things.
Once you've created a decent system for managing a lot of documents, you could try integrating it with search, which needs to be configured both centrally and locally in your site. This will teach you many aspects of the administrative process. Later projects to try are a basic publishing site and a forms filling/handling/return system, perhaps using InfoPath.
Always be alert for ways to do things in an ideomatically sound manner - a very common error when working with SharePoint is to try to force it to do something it wasn't meant to do. You should learn to write Web Parts / Application Pages / Event Handlers / Workflows eventually, but the less custom code you need to write, the smaller the chances that something will break unexpectedly.
Practically speaking, you could do a standalone server installation and install SharePoint Designer, Microsoft Office and Visual Studio on the server itself. I recommend you use a dedicated machine for this. Make sure you install the cumulative updates so at least some of the bugs are weeded out. For more realistic testing, you should network the development machine with your desktop environment and test with Office against SharePoint from your desktop.