On skills I received, during live path rolls, a +2 to any three military field skills and another that is a +2 to three basic training skills, is there a list of what these skills are related to these groupings?
Can I use these to start my mech gunnery and mech pilot skills?
There's a list of what skills are in what field in the "Important Resource Links" section of the character creation PDF. "All Occupational Fields".
When you get a field, you get all the skills in that field, and you gain +3 to each of those skills. Basic Training is a field, so when you get +2 to 3 BT skills, you can choose 3 of the skills in that field and add +2 to them. When you gain +2 to any 3 military field skills, you can choose any military field which you already possess or which you access during that stage, and add +2 to those.
What that means is that if you have not previously recieved - or you don't recieve - a Military Field which includes Pilot/Mech or Mech Gunnery skills during the Stage where you get "+2 to any three military field skills", then you cannot use those +2s to boost Piloting or Gunnery
NOTE THAT SUB-PATHS ARE PART OF THE STAGE. If you're at a military academy, you will have to go through an AIT sub-path, which is where you'll get a MechWarrior Occupational Field package. If your "+2 to any three military field skills" occurs from a Stage 3 static bonus or a Stage 3 event roll, you can still use them on skills in the MechWarrior Occupational Field.