Is there a good way to hire a better coaching staff? What I mean by that is should I hire the best possible staff in one swing, or should I upgrade halfway or so, and then do the rest at another point?

Great! Thanks for the tip!

projectrracing pretty much nailed it. The length between upgrades is up to you, but there's no need to go to step 2 if you're still training all you players in fitness. And I wouldn't got to step 2 just because you have a couple of players done with fitness.
Also be aware that when you add 20 defense and 20 attack coaches as suggested in step 2, that you also will need to buy the 20 fitness again. Same for adding 20 tactics in step 3. You buy all 4 types at 20.
When you buy coaches, you aren't adding to what you have, you're firing what you have and replacing them with what you buy.

You probably need to focus on your stadium first though otherwise youll get crushed by the wage bills