Consultants are key. You want your consultants as your GPs.
When you get training, sacrifice a consultant and whatever dr you have with the most specialties. Training is also where attention to detail (that bar with the magnifying glass) is really important, as well as with GP's and other Diagnoses physicians.
Train your DR with a specialty up to consultant and only hire low ranked DRs with other specialties. The idea here is that your new consultant will have a specialty that they can train your new hires with. So assume you train a JR with Pscych to a consultant. Then he can train a jr surgeon into a consultant psych/surgeon. Researchers are important as well, but prioritize your other DRs because a couple researchers will do their job regardless of their ability, it's just a matter of how quickly they will do it.
Now you will hopefully eventually have a high attention to detail consultant researcher/psych/surgeon.
Hire only JRs. Mass hire and fire those with no attention to detail (less than half) attention to detail means they learn faster. This is largely a patience game...but when you have a staff of about 8 consultant researcher/psych/surgeons, you can run an entire hospital with very few staff...
Aside from that. There are time when it is appropriate to "close for renovations" Since initially, the hospital should be built kind of in a piece-meal fashion, to keep it compact. Set the game speed to 1 and send EVERYBODY home. FIgure out what parts of your hospital need to be more efficient and rebuild them after gutting them completely. Short term this will cost you some money but if your hospital is big enough to need this cash flow shouldn't be a problem. Take out a loan if you need to. Remember that if you aren't overstaffed, your clinic rooms don't need to be anything special as doctors won't be stuck in them very long. If you are overstaffed, then Dr.s won't be called to other rooms very frequently.
Look through your cure statistics. Each level seems to have a prevalence of certain illnesses. Build identical rooms next to each other and have centralized seating sections.
Sometimes it's helpful to re-queue everyone in an area by sending them all to reception. They will naturally return and sit closer to the room they need to attend. Otherwise they will pick the closest seating option, which could be across the hospital.
Many washrooms, make them small-ish (3 toilets max) to reduce travel time. Also multiple staff rooms and fatigue to only 40%
It's also a good idea to manually remove doctors from rooms if you see them sitting there and there's no expected queue.
Room efficiency.
A GP will always get up from his seat on the same side. Place your file cabinet facing the seat on that side, or, directly behind the seat. Place the patient's chair right next to the room...pay very close attention to all the room animations and set them up so that your staff and patients have to travel as little as possible.
Set your policy to increase/decrease the percentage of the diagnosis process, this way you keep your patients away from you GP offices a bit longer and moving throughout the hospital. Though you will need to keep an eye out for terminal patients (if you don't stand a chance, send them to research or send them to another hospital/home)
I can't think of anything else off the top of my head.