You might want to pick up a couple of 8 foot long 2x8 or 2x10 boards from a home center store. Cut it into 2' or 3' pieces and use them to level out any angles in the ramps at ground level.
While U-Haul is the most common trailer renting place, I have found the local store pretty tough to deal with. They don't want to let a car trailer out for more than 1 day if you are staying local (1-way is OK). Earlier this year U-Haul decided to specifically refuse any type of trailer rentals to owners of Ford Explorers.
I was even told once that they would not rent me a car trailer to tow behind my 1/2 ton Silverado Z71 with a Class III hitch and factory towing package to haul my 87 Vette. No specific reason why, but they just said their trailer would not work with Vettes.
Just make sure the car is strapped down tightly. The front webbing on the tires is OK but try to get two straps for the rear. I would not use any part of the Corvette rear suspension for attaching straps. Ask for another set of wheel straps or attach a ratcheting strap to the frame or rear crossmember. Check the straps before you drive off and if your driving any distance, after a few miles. The car's suspension should settle a bit and the straps may loosen up a little.
You didn't say how far you're going or what sort of tow rig you are using, but make sure the towing vehicle is in good condition. Good brakes, check the trans fluid level (if an auto) and make sure the cooling system is working fine. Also check tire pressures to make sure they are at the pressure recommended for towing or heavy loads. Try not to carry any more weight in the tow vehicle than necessary. When you tow the trailer, you are effectively adding up to 500 lbs of weight at the back bumper (trailer tongue) in addition to the total towed weight. So try not to take the entire family or a bunch of friends. That just adds to the total gross weight of the tow vehicle. You can overload a 1/2 ton pickup very quickly!!