mattdm is always a hard act to follow. I agree with him that most decisions are artistic ones, and depend on your own personal style and vision, and to some extent the genre of photography, whether it's landscape or portraits, commercial or non-commercial.
Before you start, you need to have some idea of what you want your image to look like. High key or low key? Sharp and contrasty, or light and ethereal? Every image is different, and so no set rules apply to every image. But I will try to give some general guidelines.
Using Lightroom or Camera Raw, I work through the sliders top to bottom.
I would say in most instances this is an artistic decision. Yes, you can use the white balance dropper tool to estimate the "correct" white balance, but that often sucks the warmth out of a nice image. I do often use the white balance tool to check how it affects the image, and to keep myself honest so I don't overdo any warming, but for the most part I increase warmth until I've overdone it, then back it off a bit.
I never touch contrast in the Raw Conversion stage. I use Photoshop and prefer to apply curves adjustment layers which I can mask/brush in where needed. If I were going to process only through Lightroom, I would probably skip over the Contrast slider and return to it after making the other exposure-related adjustments.
Exposure, Highlights, Shadows, Blacks, Whites
For most images I will want a full range of tones from black to white. That is, for most images I want the histogram to extend from the left to the right. Assuming I am trying to achieve such an image, my thinking would be something like this:
Is the overall exposure roughly correct?
If the image is too dark, I'll increase exposure. Too light, will reduce it. Obviously not to achieve a nice looking histogram, but a nice looking image. A high-key image may be well to the right for example. I do try to nail the exposure in-camera, but there are times when you can't, or you intentionally expose to the right. Also if anything is clipped at either end I may adjust overall exposure. Usually I don't need to touch the Exposure slider.
Next, how are the endpoints of my histogram?
If either blacks or whites are still clipped after adjusting exposure, or more commonly if they don't extend to the edges of the histogram (meaning I don't have pure blacks or whites), then I will use the Blacks/Whites sliders to extend the histogram until I have the full tonal range. Obviously this is somewhat an artistic decision, but for those images where you want a full tonal range, this is I think the best way to achieve it.
How do the highlights and shadows look?
I sparingly use the Highlights and Shadows sliders to fine tune light and dark areas. This isn't needed at all on most images. I would increase the Shadows slider for example on a backlit image to lighten the subject, and I might decrease the highlights slider if I had a large expanse of white clouds and wanted to bring out the contrast a bit.
Increasing clarity can give the impression of better contrast and detail. Lowering clarity can give a nice dreamy look, to water or clouds especially. I think this is purely and artistic choice. I usually do not make clarity adjustments in the RAW converter, and leave this until later, in Photoshop, where I may use sharpening, local curves adjustments, or effects filters like Topaz Detail/Adjust to achieve similar effects locally.
I rarely use saturation or vibrance in Camera Raw. If the colour in an image looks flat, I may bump up vibrance, but usually I will do that to selective areas, or to selective colours, later in Photoshop. Another artistic choice - do what feels right. Just take care not to clip any channels. Vibrance is a bit safer choice.