We usually travel about an hour to the local(ish) airfield days, 1 1/2 hours to some and have done 2 1/2 hours from Bristol to Goodwood without any problems.
I wouldn't advise going on the track with your tools and stuff even if it is strapped down or if they will even let you. All organised track days that I have been on are friendly and without a great deal of people in attendance. The majority are like minded and will all have a pile of stuff taken out of their cars and left where they park up. I've never had anyone steal anything or seen anyone interfering with anyones stuff. We take a car full of stuff, 4 spare wheels, two boxes of spares, toolbox, oil, trolley jack, compressor, etc etc. Make sure its all strapped down for the journey, nothing like slamming your brakes on and the load shifting and cutting you in half when it comes to rest on your harness straps. Also I wouldn't fancy a wheel hitting me in the back.
I have lent a few bits to people when they have asked and also been offered assistance if I looked like I needed a hand, they are generally very friendly days with a good atmosphere.
Toyo Proxies are a good choice of tyre the T1-r's are good in the wet and dry and would be my choice of budget tyre. Track tyres like the Toyo R888 are also very very good but you can have good fun on the normal road tyres.
Upgrading your suspension will be a worthwhile exercise as well as your brake discs and pads, but a new set of OEM discs and pads will be fine if your budget doesn't stretch to the upgraded ones. Make sure you have decent new brake fluid with a high boiling point. Lots to choose from I use the Racing Blue fluid from Euro Car parts and never had any problems with that.
Best bit is getting out there and having fun, it can become very addictive.
Have fun
Paul