I differed slightly on this.
1. For Goals I had them on whatever they will need to get their desired classes. Though casters that aren't going Gremory will not switch into riding until they have unlocked a few spells from the opposite school of magic. On first playthrough, there is no need to set Authority as a Goal, while also instructing, as doing so will raise the skill faster then you will get higher rank battalions to equip.
-Note: there are 3 different Battalions that have stride E,D,B. The E version has bad stats so it is best to get a higher rank one. The B rank one has the best overall stats if avoid is a primary concern.
2. I focused Byleth's training on
1. Authority
2. Heavy Armor (C) / Weight -3
3. Axe (C) / Brigand Class - Death Blow
4. Preferred Weapon
This was to maximize the ability to instruct with the +2 Professor Expertise.
3. For Instructing it is much the same. Focus first on Authority whenever I have Expertise, followed by Heavy Armor and Axe to get C for getting those 2 abilities on those that want/need them. Then fallback on whatever the unit needs preferring anything that has expertise.