This one is easy. It sounds like your transmission is having a fluid problem. When was the last time you checked or changed the tranny fluid? If you’ve had it in the transmission for more than 36,000 miles (some say 30,000, the choice is yours), then it is time for a change. That change includes a new filter (be sure to check any screens as well).
There are two ways to handle fluid-changing, the right way and the fast way. The right was involves finding the transmission fluid access stop bolt. It is a slightly magnetized bolt on the bottom of the transmission pan. Open it and let it drain thoroughly. When it has finished draining, you’ll have to remove the pan cover (there are anywhere from eight to 14 bolts holding it in place) so that you can drain out the last quart or so of fluid. Be sure you have a bucket underneath the drain to catch the dirty fluid remainder. With the pan cover off, find and replace the transmission fluid filter.
Once it has drain completely, reline the pan cover with the proper gasketing material — check either the service manual or service help you can find on the internet. (Like as not you will find this whole procedure detailed on YouTube. Such help tutorials are very good.) If possible, I would recommend using an RTV-like silicon seal as it handles uneven surfaces quite well and holds up for years (RTV stands for Room-Temperature Vulcanizing. It’s an opaque or white silicon-based sealer that you can find at any hardware outlet.)
With the last of the bolts in and torqued down. You have to replace the fluid, which is easy. Most transmissions handle between four and six quarts of fluid. Be sure you refill with only the amount needed as overfilling can cause problems you can’t forsee. (Overfilling leads to overpressure damage to such devices as the torque converter, turbine blades, clutches/bands, gearing and possible either shaft (input or output which can be forced out of true).)
Check your work by running the engine for about 20 minutes and, with parking brake set and you foot on the brake — needed because of the neutral-start switch — run through the gears with your Z28 parked. It should slip into each gear easily, including reverse.
The fast way to handle changing fluid is with a high-pressure flush like those used at the discount oil change stores. It certainly is quick but the high-pressure flush can break off deposits and slam them into various parts, causing damage.