I've only done a few track days myself, but I'll add a few things I was told when I started (and a few things I wish I'd been told)
I always thought you'd have to be trying really, really hard to spin off or get a slide going, but this is frequently not the case. On cold tyres in poor conditions it can be very very easy to spin, to the extent that I almost spun on the sighting lap at goodwood! So make sure you start slowly and build up, especially as some officials take a very dim view of people spinning/drifting/hitting cones
I also thought that as I was an inexperienced driver in a relatively slow car, everyone would be going much quicker than me. This is very unlikely to be the case - you'll get some people much faster, some people much slower, and some people at about your pace. So, make sure you move over for the faster cars on the straights, but equally don't be terrified of everyone.
With regards to car maintenance, there really isn't anything special you need to do. Obviously check your tyre pressures, and make sure you've got decent brakes etc, but other than that you'll be fine. Make sure you use the cooling down lap at the end of each session to stay off the brakes and let them cool down, and avoid putting the handbrake on for a while until they cool down.
Safety-wise there's really nothing to worry about: you'll get a comprehensive safety briefing where they tell you all the rules, and they're always very quick to pull you up if they think you're driving in an unsafe manner.
Tuition is normally something people recommend: I was going to get it at goodwood but my (considerably more experienced) friend tried it first and the guy was an utter ****, so I didn't bother in the end. It's definitely something I'd like to do though, as I'm sure the majority of instructors are actually very good.
That's about all I can think of right now, have fun!
