The most important staff member for a new team is the Trainer. If you want to train your players (and you should), buy 3 or 6 young (18yo) cheap players and train them at one or two positional training. In order to get the most out of training, you need a good trainer, and at the beginning trainer level 5 is ideal. If you can't get one, level 4 will also be good (5 trains at 100 %, 4 at 97 %, so not a big drop off).
Other staff, I personally have PR manager and Doctor at 4 since they are cheap, but I know that many people have PR manager at level 1. If you are winning, your team will sell out the arena.
Other staff members, like the Nutritionist, you don't need to buy them at the beginning. Wait till you get a grasp of the game, soon you will know what they do and do you need them. Don't hurry with spending money.
Invest in Trainer and expand the arena as soon as possible and you will be set to success.