Rely on the Plug-Ins

The first and most obvious step in your path to better computer art skills is to learn your paint program (Photoshop, or Paint Shop Pro, or whatever). Take some time to experiment around with each plug-in or effect that you can generate - knowing exactly what each thing does can help you later. Along those same lines, you should be very familiar with your art program's interface, in general - learn how to arrange layers, how to select, how to path and mask things, and every other feature your paint program offers.

As an example - one of the most powerful abilities of today's paint programs is the ability to create "shadow" and "highlight" layers. Create one layer that darkens anything under it, and create a second layer that lightens anything under it. Sandwich your flat image between these two layers, then airbrush onto the dark or light layers to create highlights and shadows on your original image. This is much easier than trying to manually highlight or shadow your image. (Note: most paint programs also have "dodge" and "burn" tools, which work like an airbrush to brighten or darken an image)

But in general, experiment! It's quite possible that you can stumble upon a neat effect, and can then use that to create some neat art.

Use Simple Styles

Another great way to hide your not-so-great art skills is by creating a unique style which (coincidentally!) isn't so art intensive. Just because everyone else is doing air-brushed characters doesn't mean you have to. How about using "sketchy" or cartoon style characters? Or how about doing your game in black and white?

A word of warning: the hardest thing about doing a game in a particular style is keeping that style throughout your game. Styles work best when they're consistent - having more than one style on screen at the same time could ruin the aesthetics of your game. So, drawing background art in a style different from your character art or GUI is probably not a good idea (however, if you wanted to switch styles between levels or worlds, that would be OK).

Art Sources