I'm currently on the beginning of Chapter 6 (Blue Lions) of my first run-through of the game. I'm on normal because I usually like to get a feel for the game first before I progress to harder difficulties. My professor level is at C+ and I've recruited Shamir, Catherine, Cyril, Linhardt, Dorothea, and Ignatz. I'm currently at a B support with Lorenz. As soon as I get my magic stat, lance, and riding levels up, I'll be recruiting, Marianne, Lysithea, and Leonie. I think Lorenz will ask to join my class either this month or the next, which I'm hopeful of since I heard his paralogue gives amazing items. I don't think I'll be able to recruit Raphael, Hilda, or Caspar. Ferdinand and Bernadetta are likely to spontaneously join up if I get their supports up enough, but I don't have much hope for that since I did not invest in bows or heavy armor at all.
So far, my Byleth has at least C supported every recruitable student (no house leaders and retainers obviously), except Caspar, Hilda, and Raphael. I have B supports with Annette, Linhardt and Lorenz. I don't have any supports with any of the faculty yet, though I am sure I am close to a C support level with some of them due to all the lost items I found. I've done all available missions except for one battle mission, which I think was a quest given by Seteth.
Without revealing any spoilers, what should I be on the lookout for? I heard that some supports are locked permanently if you don't get them up to a certain level before the time skip.
I am aware that...
...so I think I have enough time to correct any issues. Again, refrain from story spoilers as much as possible.