I guess there is no right or wrong answer here.
Basically I would say: everyone should do whatever you feel more confident with - either follow the checklist or not.
Me, personally, I do from memory:
- Start
- Taxi
- After take off
- App + landing
I always always stick to the checklist for the before take-off since I had an incident years ago because I skipped it.
Nothing happened, aside from the initial: "what the hell is going on".
I was cruising in a C172 and did some testing of the AP. Once the first leg was over and I was ready to go back home I forgot to check the trim in its T/O position. It was in a all forward nose down position. I was assuming a normal rotation and climb but obviously it was not.
Thankfully right 2 seconds after pulling the yoke I realised it was way too hard and noticed the trim. Nothing happened.
After that I never do from memory the before take off one. Which includes the trim re-check (as well as the before start).
I always follow the securing the airplane too, just to make sure I do not leave the MAS/ALT and standby BAT ON :-)
If I am flying a new aircraft, there's no way I always always follow the checklist.
Regarding instruction, I am all forward to have instructors to get you used to use follow checklist.
I think following a reasonable flow makes you safer up in the air, or at least the instructor can be sure that you are not going to avoid any important check, specially when the student is soloing.
TL;DR: If you have already passed your checkride: do whatever you feel safer with.