Hi Matt, you already have good advice from Jeff, Dave, Michael and others, but I hope to add something useful to the discussion.
Wire wheels...As pointed out, many very heavy and powerful cars, Astons, Maseratis, Jags, Ferraris and even early Cobras have run wires without problems. The loads put into wheels, suspension components and brakes will be a function of the amount of grip the tires provide. Vintage race tires from Dunlop (L and M series) or Hoosier do not, in my experience provide the grip of a modern R compound track tire. I did 7 years of Solo 1 and track days on 72 spoke wires with Yokohama 032 tires with no problems; also left the brakes stock other than removing the dust shields, different pads and linings, and braided flex hoses. Only on the most severe stop-and-go layout did I have any fade.
Minilites are lighter, don't require heavy inner tubes, and are strong. If you can budget for a set of tires for track use on a set of these with splined hubs that would be ideal, and you might want to use your street tires if the track is wet, as the race tires will be shaved or moulded with very little tread depth. If this is not possible the 032 in hard compound can be driven to the track, are street legal, and have deep grooves if you don't shave them. The extra grip they offer will gradually be lost as they harden through having so many heat cycles.
Track days are a fantastic experience, you will learn a huge amount about the car and will understand and appreciate vintage racing much more, just try to keep the adrenaline under control!
Simon.