Here's what I covered in my recent initial complex checkout, in a 177RG. It could have been done in two lessons, but was done in three because we damaged a tire during an unplanned aborted takeoff.
Pre-lesson:
Exhaustive "ground review sheet" (4 pages).
Thorough reading of POH.
Checklist writing (that's my habit, not something the CFI asked for)
Lesson 1 -- local:
Preflight procedures -- especially gear inspections
Injected engine start procedures
CS prop -- firewall for takeoff, 2500 or below after 1000 feet, firewall for landing when the tach drops below setting; cycle during run-up
Cowl flaps -- when to use and when not to use (CHT gauge)
Order of configuration changes -- increased power: mixture, prop, throttle; decreased power: throttle, prop, mixture
Gear -- GUMPS at least twice during pattern, verify gear down visually (with mirror and looking out the window), and using the green light
Gear warning -- conditions that set it off and don't set it off (177RG does not set it off with flaps, but does with throttle); ALWAYS drop the gear when you hear it unless there is a preplanned reason not to.
Slow flight & stalls
Normal takeoff and landing
Lesson 2 -- nearby airport
Departure and arrival procedures
Leaning
Cruise performance
Descent -- slowing with gear and flaps (subject to V-speed limitations); gradual throttle reductions
No-flap landing and takeoff (aborted)
Lesson 3 -- different nearby airport with a longer runway
Departure and arrival procedures
No-flap takeoff
Distractions -- unfamiliar crossing under Class B between two remote airports (this was a sort of "final exam")
Soft & short field takeoffs and landings
Most of this was just a checkout in a new aircraft. But, the instructor wanted a whole bunch of landings to make sure I wasn't going to forget the gear. It didn't matter too much what kind of landing they were after each type had been demonstrated adequately. Total Hobbs time was just over three hours. It might have been longer, but I already had an HP endorsement and some experience in a 182Q, which has a CS prop and cowl flaps. The "rule of thumb" at the club is 5 hours, but the instructor's opinion trumps that.