I come from the land of game programming and a very logically oriented mind with a bit of a dulled experience. I have experience manipulating images in Photoshop, doing tasks like slicing and tracing but nothing like creating things from scratch.
Where can someone with my sort of background learn to create vector (or pixel) art?
Learning to use any app is the same. Get a copy and start experimenting. Create some projects for yourself or go find a tutorial and off you go. VectorTuts is as good a source as any for tutorial inspiration.
Learning to do something with an app is a different question. You used the term art which implies doing something more than a simple website wireframe or basic layout. You need drawing skills. I've provided my thoughts on that process before. I highly recommend the skill for any visual or conceptual pursuits.
Manipulating images in photoshop is still pixel art. Learning illustrator is just like learning any other drawing software. I'd recommend diving into the program with a book, or signing up for lynda.com and checking out some of their introductory tutorials on illustrator.
If it's the actual process of knowing what to draw and how to draw it, not simply about the software, then what you're looking for is how to draw. This I can't really help you out with. There are all sorts of formal drawing classes out there you can sign up for. A lot of people just enjoyed drawing as a kid so they did it more, and grew better and better at it growing up.
Vector and pixel art are completely different things. Your question mentions both.
Like any art, you would learn from the ground up regardless of your background. Art is art. If a 3 year old can create art you can. There's not really much to learn to "create art". The learning curve comes in when you want to create artwork for a specific purpose.
You didn't go into any detail. Your question is a bit like "can I learn to cook coming from a butcher's background." One has nothing to do with the other and learning to cook for a Michelin star restaurant is a great deal different than learning to cook for your children.
If your goal is to learn to create vector art, you would find a vector application which you enjoy using and start learning about how to use that application. The same would hold true for pixel art.
The only important aspect would be to choose the proper application based on your desired final output. For example, Photoshop does not create true vector files. Regardless of anything you find about "creating vectors in Photoshop" you can not use Photoshop if your goal is to end up with a truly vector file. You need apps such as Illustrator, Xara, CorelDraw, etc.
For pixel art, Photoshop works as does any of hundred other pixel-based editing applications.
