What order should you go through the Camera Raw editing controls?

The answer ends up being a personal workflow choice, of course but this sequence can get you started. Generally, work from Global adjustments first then Targeted adjustments.

1. Choose Color or B&W and your Profile from the Basic Panel: these settings affect your starting point with the image so they are appropriately first.
2. ACR only, not Lightroom Develop, look into the Workflow Options (this is the old style linked text at the bottom of the ACR window: If you are prepping images for a certain print size or use this can save a lot of time. If you just want highest quality and most flexible workflow then this can be your normal settings.
3. Crop and Straighten
4. Spot Removal (Unless there are a lot of them! It can slow down the editing)
5. WB & color cast corrections
6. Basics panel for exposure, contrast, and basic color adjustments
7. The targeted adjustments: adjustment brush, radial filter, graduated filter
8. Noise
9. Sharpen
10. Re-visit your Camera Raw/ Develop Preferences?