This is a huge question because "design" has two meanings: "decoration" (graphic design) and "planning" (experience/UI design). I would argue they both influence "aesthetics" (a point often lost on people with formal training in either fine arts or computer science, who sometimes see "design" as something pretty slapped on top of a stack of engineering.)
If you're interested in improving your graphic design chops, you'll need an understanding and appreciation of basic design principles: contrast, proximity, color and typography. I also wouldn't hire a designer who can't draw perspective, still life, and human figures on a piece of paper with a pencil or charcoal, and without using rulers. I believe aesthetic and artistic ability can be learned (others argue aesthetic judgment is to some degree inborn). But at a certain level, you'll need to care a lot about how things look. (Litmus test: every designer in the world has a passionate opinion about Helvetica. Do you?)
All the good graphic designers I know spend some of their free time design doodling: a fun game is to take a piece of bad design (like a used-car ad or a Thai takeout menu) and make it into a piece of good design, but without altering the copy.
For graphic design, I'd recommend starting with a few books in design basics:
- The Non-Designer's Design Book
- The Elements of Typographic Style
- Designer's Color Manual ad/or Light and Color in Nature and Art
Books by Wucius Wong, particularly Principles of Form and Design
- Meggs' History of Graphic Design, and/or Graphic Design History by Steven Heller
- The works of Edward Tufte
Taking a few classes in design basics and drawing at your local art school or community college wouldn't hurt.
I'd be hesitant to hire a graphic designer who couldn't wield Photoshop and Illustrator with proficiency, although someone with a good eye can learn these apps on the job.
I question whether experience design can be profitably learned from a book or class. It comes from working with all the parties who interact with a website: users, graphic designers, product managers, developers. IME really good experience designers begin with business objectives and then work with users or test groups to develop a mental model for delivering those objectives. They then break that model down into task domains (which might map to features) ... each of which gets its own user experience flow, wireframes and so on.
I guess I'd recommend these books:
- Information Architecture (Rosenfeld)
- Designing Interfaces (Tidwell)
- Be familiar with Jakob Nielsen's work but don't let it run your life
- Edward Tufte (again)
There's no standard technologies for UX design, although you'll probably use Visio or OmniGraffle at some point. Study other designers' wireframes and flowcharts. Michael Angeles and Khoi Vinh have a lot of their thinking online.
In regards to web design in particular, the future seems to be frameworks and modular CMSes, with a rich interaction layer (ie. AJAX). Build a website in Drupal, Django, or Rails, just to see how those things work, and add a few Ajaxy elements to get a feel for those, too.
Finally, spend a lot of time on the web and pay attention. Think hard about every web experience you have. Good UX is invisible: It makes tasks so simple you don't have to think about them. So every time you find yourself trying to figure something out on a website, ask yourself how you'd make it better.