One of the easiest things you can do to get a little more out of your car is to make sure your wheel alignment is where you want it. It should cost about $200 to have an alignment shop set you up.
At least for standard issue Corvettes, the factory alignment settings are optimized to reduce noise, vibration, and harshness, and to maximize your fuel economy and tire life. They result in stable, confident steering, but there's a tradeoff - your car has increased rolling resistance and it doesn't turn as well as it might. This is critical stuff for autocrossers, but you can feel the difference in any venue.
All production cars are factory-specified with a little bit of front wheel toe-in - that means that the forward faces of the tires are slightly (about 1/8th of an inch in most cases) closer together than the rear faces. Similarly, most production car rear wheels are designed with some amount of toe-in. The practical effect of this setting is that your wheels are always driving a little bit towards the center of the car. This gives your car a nice stable feeling on the road, but it also means you have increased the rolling resistance of your tires. Toe-in also means you have to turn the steering wheel farther to get both front tires pointed into a turn, and the inside tire won't ever be pointed as far into the turn as the outside tire.
To maximize handling on an older (1963-1982) Corvette, you can ask an alignment shop to give you about 1/16th to 1/8th of an inch of toe-out in the front and to set your rear tires up with zero toe to 1/16th of an inch of toe-in. First-generation Corvettes can set front toe, but not rear toe because they use a solid rear axle.
Setting toe-out means that the front wheels will be driving away from the car's centerline, making your Corvette more eager to turn, and the rear wheels will be exactly parallel to reduce rolling resistance - but the tradeoff for quicker steering is reduced stability on the road. For maximum straight-line speed, set your car up with zero toe all the way around.
If you have a 1984 or later C4-5-6 Corvette, you will want to specify zero to 1/16th inch of toe-in all the way around.
The other alignment settings you can adjust on your Corvette are camber and caster (also spelled Castor). Simply put, camber is how your wheels are tipped-in (or out) at the top, and caster has to do with your steering axis. Don't worry about changing your caster away from stock settings, but you want to ask the alignment shop for ½ to 1 degree of negative camber on your front wheels, and ½ to ¾ of a degree in the rear for 1963-1982 Corvettes. Again because of that solid axle, 1953-1962 Corvettes have no rear adjustment. For newer Corvettes from 1984 onwards, you'll just want zero camber (straight up and down) to about ¼ degree of negative.
Remember - any time you use lowering springs or other suspension changes, you need to have your car re-aligned.
So, we know that alignment is important, but if you've just put out the money to align a set of worn-out tires, chances are you won't see a lot of result. So consider whether you could use a new set of wheels and tires. Everyone knows that good tires are critical to good performance, but your wheels are also an important performance component because they are unsprung weight. Your wheels affect your acceleration, braking, and suspension response.
The stock wheels that Chevy gives you are designed to survive all kinds of abuse and keep going. They're very heavy. You can find substantially lighter wheels in correct sizes for your 'Vette through the aftermarket, but be ready to pay up to several hundred dollars apiece for them!
If you plan to do your performance driving using special tires, you should buy an extra set of wheels and keep your competition rubber on them. Autocrossers, racers and track day enthusiasts should all have a dedicated set of doughnuts kept in good condition and reserved for playtime.