"RJ" and "Dad",
Welcome to the
OldMan32, you're a pretty awesome dad to get your son a Corvette for his first car! Wow...
Shoot me a PM with your mailing address and I'll send you out a copy of the 1980 factory service manual on CD. I have a couple extras laying around.
Generally speaking, without knowing more about the car (mileage, engine, transmission, pre-existing conditions), all the things that you'd check on any used car apply here. In this light, "It's just a car." So check out all the usual things, starting with the safety components: brakes, seat belts, etc. If the car still has the OEM brakes, you'll probably be changing the calipers to the newer style. Then focus on the fluids. Are they all good and flowing as expected? Are the gauges reading correctly?
After that, you can move on to the mechanical stuff. U-joints and bearings seem to be a fairly common replacement on the late C3's.
Troubleshooting 1980's and earlier is pretty simple. You've got fuel delivery, vacuum, and electrical/grounding. That's pretty much the big three that will keep your car from running properly. If you get those three things down, your car should be pretty much 'by the book' for everything else.
While I don't discourage posting new questions when you have them, keep these two things in mind.
1) These cars have been around 30+ years. If your car has symptoms/problems, I'm fairly certain you're not the first, and thus it's probably been discussed in these forums. Learn to use the
Advanced Search functionality of the site...make it your friend!
2) There are lots of members here and we have people of all ages and walks of life. Some of these people actually worked the line for Corvette production or had a piece in creating the Corvette over the years. The combined knowledge here is unsurpassed, as is our willingness to help. That being said, we don't live here. If you pose a question and don't get a reply within 30 minutes, don't repost it elsewhere on the site nor do a "bump to the top", unless it's been days/weeks with no response. Typically,
someone will chime in with a response which will either help you or at minimum keep your post visible to others. In other words, "Be Patient!". Your car is a long term project...you've got time!
Was the picture of your car taken in your yard or the previous owners? It looks like there's another C3 parts car in the background.