It depends somewhat on what level you are in. In D.IV, your finances will be pretty tight. Ordinarily, you will lose money each week, and then make a lot of money for the TV games. New teams get $50,000 bonus each of the 1st 4 weeks, so make sure you're not considering that as regular income.
People aren't really sure of the effect of the PR man, so I wouldn't upgrade that.
Doctors lower the likelihood of injuries and their severity. They don't make injuries heal faster. So you can't wait until you have injuries and then upgrade. They are more like an insurance policy. So you might consider the value of what you are insuring. If you don't have high value players or high value trainees, it might not be worth spending too much on a doctor. If you had a NT player, you'd want a 10. I'd probably get rid of Max Payne, but wouldn't go too high.
The trainer will speed development of players, but won't have any effect on games. You probably can't afford a 10. So perhaps a mid-range if they are available.
The staff system is kind of odd in BB. Each week there are 3 staff members available, one of each kind. So if you are looking for a mid-range trainer, and there is a 4 this week, you might pick him, because next week it might be a 10 (too expensive, or a 1 too poor), and then keep watching for somone more to your liking. You will pay an extra week's salary when you switch a staff member.