Those are the absolute worst vehicles imaginable in terms of efficiency and maintenance in stock form. They also drive like garbage in stock form (very flexy body, garbage suspension, narrowish tires with huge sidewall, slower than a modern i4 econobox, etc).
Now, even though I said all that, not all is lost. If this isn't a car you're going to care about restoring to all-original matching numbers kind of status, then you can actually build a nice little C3 *fairly* affordably if you do a lot of the work yourself.
Step 0 : Research all this and make sure you have the time/money/patience for this.
Step 1 : Find one that hasn't been mangled.
Step 2 : Get a crate LS1 (or warrantied used one from a reputable vendor)
Step 3 : Get a T56 tranny, probably from the same vendor.
Step 4 : Get all the associated gear, wiring harness, mount adapters, gauges, BRAKES, etc. As a bonus, you may be able to skip A/C entirely depending on where in Cali you're moving. You will want to swap the rear end out as well of course.
Step 5 : Work, work, and more work. And probably a few parts/etc that you hadn't considered at the beginning of the project. You may want to get some new but classic-looking wheels and some better shoes for the guy. If you want to go high-HP, then you must look at stiffening the chassis up a bit to avoid twisting. You may want to look at aftermarket suspension setups that will give a better ride and handling (as well as easier parts to find in the modern era).
Done right, you'll have a bulletproof (LS1 + T56 = solid as it gets) car, built fresh from the chassis up, that you'll have a lot of fun with and a personal story behind. It's not super easy, but it's far better than those old carb'ed jobs in every aspect (perhaps other than the sound of a really well built one). Emissions, reliaiblity, fuel economy, weight (!!), HP, torque, availability of parts, etc, etc.
You *could* deal with a stock L81 or similar, but gah. It's no fun getting 13mpg and getting stomped by a Camry in both handling and acceleration.
For the fun of it, check the MT review (this was reposted online in '03, but is the original review of the 1980 model same gen as 81)
http://www.roadandtrack.com/car-reviews/1980-chevrolet-corvette
3 and 4 speed transmissions.