I'm only about 25 HPDE days into this, but here's my experience.
There is nothing "fancy" I could have done to my car that would have decreased my lap times for about the first 15 days, or speeded up my learning. It took me that long just to learn how to maintain concentration for an entire session, turn in at the right spot and right speed, hit all my apexes, and track out to the right place consistently. It's a vision and feel thing.
I have been doing this in one light 140HP old BMW and one heavy new 400HP BMW. I learn everything faster in the light old car.
If you are driving a heavy-ish car on bigger tracks, somewhere in the first 15 days, it will be time to get heat-tolerant brake pads. There are pads out there like StopTechs that can take the heat of the track, don't eat your rotors, and don't cost an arm and a leg.
At about 20 days I evaluated the situation on car changes. I went with stickier tires, because that let me level up my learning on braking and cornering at higher speeds. This was on the advice of my instructors at the time; they wouldn't have wanted me on sticky tires before then. And in-car video, because I found I figured out a lot looking at video a couple days later.
I read Fred Puhn's excellent book on racing suspension, and studied it until it all made sense. But I didn't change anything in my car because I frankly wouldn't have been a good enough driver yet to tell whether it was helping or hurting
At about 25 days, I moved the rear spring perches on the old light car up one click to reduce neg camber and increase travel a bit.
So far, my best investment of time and money by far has been HPDE weekends with good in-car instruction. No question. I'm not sure how much autocross helps. The runs are so fast and it's all about transition from moment to moment, which isn't what I do on track. I'm sure it doesn't hurt, but plain-old wet skidpad seems to be a more transferrable car control clinic for the track.