I've done a search, but found no thread of this type already.
So, here's some information about Friendship, triggers, and effects. Please contribute as much as possible.
Note: I haven't much accurate information on my own, as I have started to pay attention very late to this matter. I have since restarted the game and try to record the triggers more accurately this time.
Random <-> Guanidia:
10 Fights: PARTNERS - XP boost -
110 Fights: FRIENDS - Speed up OR Speed up Lv2 for Guanidia -
400 Fights: SOUL MATES -Random attacks after Guanidia casts a spell-
Random <-> Sarah:
The fight clock starts to count once you reach Sankt Leona.
For instance, I had 1520 fights with sarah&random as I reached Sankt leona.
1570(50) Fights: PARTNERS - Armor Up, Weapon Up Lv2, Mind Up Lv3, Speed Up Lv2 For Random in battle
1600(80) Fights: FRIENDS - Sarah learns the spell "Striker"
Random <-> Lisa:
80 Fights: PARTNERS -> Lisa learns Tactical Move
#EDIT:
Tested Guanidia/Random. Turns out SOUL MATES is achieved at EXACTLY 400 Fights together.
#EDIT No.2:
Accurate information about Random/Sarah, and some about Random/Lisa.