<<. I know how to correct by finding a spot that should be neutral and using the eyedropper>>
You have to be careful with using the neutral(gray)eyedropper. Because, if the object is not a true(gray)neutral, and you click on that object it will force it to be a neutral. Creating a cast in the rest of the image.
The way experts do it Is by doing global corrections to the image. In rgb........ setting your highlights,shadows to equal(neutral) values. Example: for Highlights 245 R,245 G,245 B. Shadows: 10R,10R,10R. "Grays" are the same as well. Example: 127R, 127G, 127B. Or 115, 115,115. See where this is going. Set your highlight and shadow first, then do your "gray", if there is one in the image. Sometimes their is no highlight or shadow(per say) in the image either. So you can set your highlight and shadow to your lightest and darkest point in the image (using the threshold command to find those points). One might say, if their are no (gray) neutrals in the image, what can one do. Well their is a technique called "known colors". Fleshtones, grass, the sky. Example: grass, has more yellow than cyan in it. The magenta is low. These cmy values can be(and are)related to rgb as well...... Using the info pallet settings...........So (using the info pallet) checking these colors, if the yellow is lower than the cyan, and the magenta is high. You correct these values(colors) to their right ratios. These colors are not neutral,btw. But the image can be correct for casts and balancing in this manner. And btw, not only do you check your image by your well calibrated monitor, but also using your(good old) info pallet(checking those highlights, shadows, neutral, and "know colors").
<would really like some tips on ways to improve my skills at assessing color problems>
Their is a book called "Professional Photoshop" by Dan(the man) Margulis, as well as the web,other books.
Hope this helps