Lots of good advice here
Ultimately, what do you want to achieve from this? Is it better control of the 350 on the road, learning how to drive on track, or just a lot of fun in a new car that's ultimately disposable? None of those options are wrong btw!
I've done a lot of track miles over the years in lots of different toys, and whilst I've leaned a lot about car control that I've transferred to the road, it's not all transferable. Older me now views younger me as a proper liability on the roads, back in the day. As an example, blasting round a treacherous Donington last month in the MR2 in the pouring rain on 888s was huge fun, but it taught me very little about driving fast on the road in the rain as there's no way I'd ever attempt to drive at pace in those conditions on the public roads. Skill transfer = zero.
On the flip side, Anglesey in the dry in my old VXR220 was a revelation. Something inside finally twigged back in 2007 and I've been very comfortable driving MR cars ever since. That made my road driving considerably quicker, as I had the confidence to push where appropriate. Skill transfer = a bucket load.
Going to a mk1 MX5 was a blast. It was my first ever track only car, although I kept it road legal to avoid trailering issues. I bought a lovely example for £900, put the crappiest tyres available on it, added bucket seat & steering wheel & gearknob, a rollover bar and that was it. Total cost was about £1500. I learnt a hell of a lot about FR balance in that, much more than I'd ever picked up in my Zed, both on road and track. You had to be more committed to get the car to rotate due to naff all power, but the crappy tyres meant that it was easy to keep in an angle. It was completely disposable, so I took liberties in it that I would never have tried in my 'decent' road cars.
I guess what I'm trying to say is that a track car can be bags of fun, and you can learn a lot, but I've spent a lot of money to get to the level I'm at. I could've saved a packet and probably been a better driver if I'd just spent the money on driver training instead, but that for me wouldn't have been as much fun as teaching myself, mistakes and everything.
If you want to learn how to handle an out of control car, go see Andy Walsh at a carlimits.com day. He's my go-to guy for driver draining, and it's a stonking day out for about £250. Four days with him will see you better than 10 trackdays. If you want to learn how to drive your Zed on an actual track, then get it insured and find a safe track to go play on (Bedford is good, as is Anglesey), and enjoy. If you want something you don't care about destroying then grab the crappiest looking MX5 you can, just make sure there's no rust and it's mechanically sound.
Btw, I've a standing offer here for any 350Z-UK member that as long as my calendar is clear and you pay for me as a second driver, I'll happily travel to any track day with you and offer tuition for the whole day. Did one back in March for a guy on here, and we both had a great day, he was getting some serious pace by the end of the day. Just another option for you.