My current understanding is that Hunter Rank increases based on story progression for a while, and then at a certain point, it "unlocks", and will increase based on any quests.
I've found a few sites that describe it at a high level, but I'd love to specifically know how it progresses even based on the earlier story quests; my regular co-op pals are a little ahead of me, and I can't tell how many quests/expeditions/etc. it will take to rank up so I can join them on quests. Being able to see how many quests between certain ranks would help me gauge how much I need to accomplish before catching up.
EDIT @Vemonus's answer is helpful; however, I'm looking for what specific quests, expeditions, etc. even within the main story which will increase my rank. I understand that there is a cap of sorts, which keeps one from grinding and skipping through the story. I also understand that "Assigned" quests are the story quests, but I've found that not every story mission seems to increase my rank.