First off, never take your car to a tire shop for an alignment. The data they have is based on old technology, old tires, and is not for performance driving. Even a spirited street car needs more attention than they will give it. An acceptable specification to them for camber may be between +2 and -1 degrees. The car will do crazy things when the alignment is far out from the desired specifications. Every car and suspension manufacturer is different, but as a baseline, figure a sport street car should have between -0.5 and -0.7 degrees negative camber, 2.5 – 3.5 degrees positive caster, and between 0.0625 and 0.250 degrees toe in. The same car set up for a road course would benefit from -2 to -3 degrees negative camber for a better footprint during extreme cornering. A street car would feel twitchy over uneven pavement and would wear tires faster with this type of setup. The caster would need to be as positive as possible to create straight-line stability and improved turn in. A street car will ride nicely with maximum caster but will experience a decreased turning radius, something that's important on the street, but not at the track. A track car will have much better turn in with 0.125 to 0.375 toe out, something that would be uncomfortable on the street and will scrub enough to actually handicap a lower-powered car's acceleration. Autocross cars get very similar suspension treatment though they are usually stiffer in the rear to promote oversteer, an important tool for getting around a tight course.
Tires are the most important change you can make to your car, so choose the right ones. Sittner's favorite tires are the Hoosier R6 for road course and the A6 for autocross. These tires may be nearly identical, but have different characteristics. The R6 is a slow-wearing and slow-warm-up tire that's designed to keep grip over long periods of track time. The A6 however, is designed to heat up quickly, but will lose its grip in a couple minutes. That makes the A6 perfect for autocross and even for qualifying at a road course where you only need a couple laps of grip. They both have a tread wear rating of 40. The lower this number, the softer the compound.
If you're planning on driving the car to the track with the same tires for the session, Kevin recommends the Nitto NTO1. It's a 200-treadwear tire that has grooves designed to tolerate water. It's a great tire to keep around for a track day that gets rained on. Many classes have tire requirements either by tread wear or vintage. Goodyear makes a vintage tire legal for nostalgic racing, but it shouldn't be used instead of a modern tire when you have the option.
To get the most performance out of whichever tire you choose, you need to cycle it once before the event. Run the tire for a lap or two then remove them from the car to cure overnight. This will extend the life of the tires. A brand-new tire will never run better than its second heat cycle, but if it's not left to cure after the first, it will wear at an increased rate.
Tire pressure is just as important as tire choice. There's no magic number that will work for every car, but there are signs. You will be able to look at wear onto the sidewall of your tire to determine if you need to add or remove pressure. If there is wear too far down the sidewall, increase pressure, if it doesn't reach the edge of the flat contact portion of the tire, decrease pressure. It will take some experimentation, but people at the track are usually quick to help out another racer if you need advice. Adjusting tire pressures can both improve the contact patch and overall grip, as well as change the oversteer/understeer balance of the vehicle.
A car with a decent amount of power is going to cross the 100-mph mark on any road course, so all of your rotating parts better be in balance. Imagine the frustration of getting your new slicks on your car and finding out on the first straight that they are horribly out of balance. Wheels are the first thing we think of to balance, but there other parts that can cause a distracting or even damaging shake. The wrong flywheel, misbalanced driveshaft, or bad rotors can wreck your track day. The most sneaky of shaking issues comes from excessive axle runout. Kevin takes all of his customers' axles and runs them on the lathe to verify their flatness. It only takes a couple minutes to remove the studs and turn the axle. Here you can see what a 0.025-inch cut off the face of an axle looks like. The most runout you should have is 0.005-inch so you can see how far off this axle was. It wasn't until 0.030-inch that the cutting bit touched the whole face.