I like that year Vette..I myself own a 1976 which isn't too much different mechanically speaking.
From my experience, go right to the brakes. Chances are they are probably shot. If you look through a search on this site you'll find a TON of advice on how to fix the brake systems. Not being a bad mechanic myself, I would give you some advice...plan on simply buying four new calipers if you suspect brake problems at all. Most repair shops will charge you an arm and a leg to try to 'fix' the problem..but the real solution is to buy new and spend your time driving and not under the car. While you are at it, change out the emergency brake system while you have the wheels off and rotors exposed. Made that mistake..and was right back at it two weeks after just replacing the calipers.
Secondly, I would look at the front steering components. Usually the power steering cyclinder and power steering control valve on these things is worn out by this time if not prevoiusly replaced. Tie rods are also another culprit, due to the wider tires on these cars.
If that hasn't scared you off (lol)...take a real close look and listen...to the rear end. As these things have IRS, the u-joints and rear spindles take a fair amount of abuse. Again, so a search o this site so as to know what to look/listen for. Obviously if you hear a "clunk" when you start to move forward or hear the same thing when you decelerate down to a cruising speed etc..you got some of those problems. To be honest, I have been driving mine with shot u-joints for quite some time...is that smart, no...but I don't race the thing either. I DO expect to be calling AAA though sometimes down the line.
Those are obviosuly SOME of the big things I would tell you to consider. The other thing specific to the Vette is the vacuum operated headlights.
Although these cars have many more systems than say the same year Camaro, I have absolutely fell in love with them and put up with their nuances for the shear pleasure to get to drive them. That being said, they are not cheap. However, to hear that small block Chevy is a complete joy!
Oh, by the way...I have about 100k miles on mine and I drive it as a daily driver in the Summer...and this is without having seroius routine maintenance done on the car for the past 20 years...the previous owner to me really let the old gal go....but I'm bringing her back, one piece at a time!