My opinions, based on my research and study of the game's workings:
- Drivers: Like Mr. Redmeijer said above, you won't find a truly decent one until about Level 4. Once you get there, hire the best one you can get. I look for a balance of high talent, young age, high health and high stamina. Talent can't be boosted by training; it's a static value. Young drivers don't atrophy like older ones (my experience says drivers will stagnate about age 29 or 30). High health and stamina means you can have them push harder through the race, which can come in handy depending on your league/tier. When your driver's health goes low during a race his/her lap times suffer. I've won races simply because my driver's times didn't fall off like the rest of the field's did and I could make up places later on.
- Chief Designers: Always try get the best one that's available (sort by Overall Level, the graphs for Experience and such don't matter). Higher level designers increase your rate of car development and also the potential performance.
- Commercial Managers: They don't do anything. You should Auto-Sign this position if anything. Marketers are useless until you can get the Commerce facility at Level 5. Fire them and shift the staff spaces to Designers, Engineers and Mechanics.
- Technical Directors: The Experience rating of Technical Directors influences the Litres per Lap estimate you get when preparing for your next race; otherwise they do nothing. Another slot to Auto-Sign.
- Doctors: I personally have found no evidence to suggest that better Doctors decrease driver atrophy rates in the game in its current state. The only thing I've found that affects atrophy rate is the age of the driver.