Hey TJ,
I've never really heard of apprenticeships, but I'm hardly well placed in web design networks. Perhaps someone with more experience would be able to point you to something.
Personally, I've always been a fan of learning by doing. The way I initially learned web design and development was to start working on projects. If you don't have anyone who wants a free site from someone who doesn't yet know what they're doing, then design a site for yourself (even if you don't want one). If you're wanting to learn web development, pick something simple that you'd like to see a site do and then create something that does it.
Even if your early projects never make it to the web (sometimes it's better if they don't
), the learning experience will be well worth it.
There are also loads of tutorials and so forth around the web about designing, and lots of great blogs and so forth that talk about good design practices and whatnot. Starting to read those just to be aware of the issues that are out there for web designers isn't a bad start.
Good luck!