(https://www.hattrick.org/en/Help/Rules/Specialists.aspx)Ones that relate to BB:
Assistant Trainer (max 2, speeds up training, increases injury risk, slight boost to form)
Doctor (reduces injury risk + length)
Form Coach (raises player form i.e the Psychologist in BB)
Ones that don't:
Psychologist (raises Team Spirit and Confidence, no comparable element in BB)
Financial Director (basically increases the max cap allowed for hoarding money)
Tactical Assistant (HT allows tactical changes during the game based on criteria set pre-game, e.g if down 2-0 then bring on another striker. Not comparable to BB)
In my opinion the Hattrick system is very basic. You can go years without thinking about your staff, just set them to a season-long contract and set them to auto-renew at the end of it. As JoeMaverick says you can hire from level 1 to level 5, but staff is so cheap that mostly people hire level 4 or level 5, so again nothing to think about.
The only thing to consider is how to use your four slots - if you want to maximise training you hire two assistant trainers + doctor. If you are competing then you might want the Psychologist and Form Coach, maybe a Tactical Assistant, leaving only one slot for an assistant trainer or doctor.