I know that Proformance Racing school will give you a Sport Driving License after attending a HPDE class and a lapping day. What that means is if you go sign up for a lapping day at Pacific Raceways with a car club, they will let you in with or without an instructor depending on how much experience you have had. It also means that you can go to other tracks and they will recognize that you've already done track work and let you sign up for track days. But they will probably require you to have an instructor with you for the first few laps to get you familiarized with the track.
Just to be clear, HPDE and lapping days are not timed. If they were timed, it will be considered a race and it raises all kinds of insurance issues for you and the track. These are all driver education events. Of course your insurance will not cover anything should anything happen during these events. And it's best you don't tell them....their job is to collect premiums from you and deny your claims whenever possible. There are outfits that sell per-day track insurance if you are interested.
It is WAY safer to drive your car on the track than drive fast on public roads. Tracks are designed with grassy run-offs in the tighter turns. And these educational events have cones to guide you where the braking zones are, with turn-in cones, apex cones and exit cones. Everyone on the track is there to learn to drive their cars fast and safe whereas on the public roads you have obstacles like fences, curbs, light posts, other drivers on the phone talking to their BFFs, texting, painting their nails and their lips.
Go to youtube and check out some video on HPDE events.
Autocross is completely different from track events. They are timed events and I suspect insurance companies will frown on them too. I have never done one but looking to do some this year. I am on the wait list for a Novice Autocross class for 3/10. These are layout on a huge flat parking lot with cones. And you go through them to get the fastest time. I imagine you would barely need 3rd gear for these events.
Of course wear and tear will be greater doing these events. You gotta pay to play, right? But the biggest expense is gas, brake pads and tires. I usually do an oil change before or after track days. I am going to tell you that the GTI motor is just waiting to be rev'd hard for the track. And the chassis is simply rewarding to drive on the track. However you will find out that the stock brake pads are not made for extended track days. I bought new performance pads after my evening track day last summer with my TDI.
You will have a lot of fun after a track day. The adrenaline high will stay with you for at least a week after your first one. You will also learn so much about your car and your skills.
I also participate in other car-related events: TSD rallies and Chump Car race series. Good luck to you and congrats on the GTI purchase. Trust me, you will love it on the track. I love driving my TDI on the track!