Personally I am a big believer in flows. That is all the checklist flow in a logical order from one item to the next. Therefore the written list is used after the flow has been carried out in order to insure that nothing was missed and the written document is a true "checklist". For Cessna's I start at the fuel selector and go in a counter clockwise circle through the engine controls, circuit breakers, engine guages, radio stack, flight instruments, switch panel and back to the engine controls. This flow works for every phase of flight and the engine failure drill. However there are numerous opinions what is the right checklist for low time pilots. Most schools IMO have needlessly long checklsit full of trivial items and treat obvious items like letting go of the starter switch with the same importance as pilot killer items like making sure the trim is set for takeoff. Anyway rant over.
Some things to think about.
1) In your simple fixed pitch prop trainer/tourer, the only way to know the engine is making full power at the start of the takeoff roll, is to know what the minimum staic RPM limit is. Any RPM below this value at the start of the takeoff run should be grounds for an immediate reject. Contrary to the assertions of one PPL poster here, it is very difficult to "feel" a underperforming engine, but the static RPM check will give you hard information.
2) Be very carefull about letting yourself get head down reading checklists when you should be looking out the windshield. This is a particularly important around the airport (eg the prelanding check).
3) Regardless of what the checklist says leave your strobes and landing light on in airspace where there are likely lots of aircraft (ie around an airport and in the practice area) it makes you much more conspicuous particualarly on hazy days.
4) When you are settled down in cruise make a note of the exact position of all the engine guages. Engines or electrical systems that are starting to die will almost always give some warning. For example an engine starting to come apart inside will show a slow but steady increase in oil temp with a simultaneous loss of oil pressure. If all you are looking for is "in the green" you will miss these warning signs.
5) Carb ice is one of the leading causes of preventable engine failures. Carb heat should go on at the first sign of a reducing engine RPM and/or engine roughness
6) Almost all in flight emergencies do not require instant action and a slow methodical approach should be taken when going through any emergency checklist. Starting with asking the question, " am I sure I understand what is happening"
7) All system emergency checklists ( eg alternator failure) should have as the lat item on the checklist the page(s) number in the POH that overs the emergency. All modern POH's have considerable extra detail on how to handle an emergency in the POH over the simple list of actions on the emergency checklist.