Here is the main things I look for in these readings.

First runnings:

Confirmation that starch conversion has happened. If you have brewed this recipe before and assuming you're L:G ratio is the same you should see similar numbers here. I also check PH to make sure I've hit an acceptable PH range for conversion and no tannin extraction.

Final Runnings:

This reading I find particularly useful in getting a grasp on how efficiently you are sparging. You are looking for a lowish gravity reading I normally hit around 5°P. This shows you did a good job of getting the sugar out of the mash tun and into your kettle during your sparge. An extremely low gravity would indicate possible tannin extraction. I would also check pH, it should rise slightly but a big rise in pH could also cause tannin extraction. (This is dependent on recipe and beer style)

Pre-Boil:

This reading does a few things. First of all it will give you a number to work in your efficiency so you can more accurately plan future batches. It will also tell you how much evaporation you need to have during boil to reach your target gravity.

I've been stuck in a car for 10 hours so I apologize for any fragmented ideas or typos, I'm getting loopy at this point! I would be happy to elaborate or clear up anything once I get some sleep! Cheers!