Welcome to the board, ljmerrill010! A DIY kid? Smart choice. You've come to the right place.
If the click you're hearing is a loud one, more of a clunk from the starter, it is commonly a problem of the bendix hitting the flywheel/flexplate and not engaging the gears.
1) One possible cause would be improper shimming of the new starter. But in that case, you would have seen this happen shortly after installing the new starter.
2) Another possible cause is electrical. If the starter solenoid is not getting full power from the key ignition circuit (bad connections in one of more of the several components in the path of the current), it can only extend the bendix to hit the teeth of the flywheel but won't have enough power to engage the teeth (at which point, at full travel, the starter motor engages and turns the engine over). You can easily check for this by putting the car on ramps, setting the parking brake, making sure the transmission is not in gear and from underneath the car, using a big metal tool (screwdriver) to touch the small starting post to the big positive post on the solenoid. That will bypass the starting circuit and will provide full power to the solenoid. If that works, you have a weak component (bad connection) in the starting circuit. That could be in the ignition cylinder, in the Starter Enable Relay, the shifter Park/Neutral switch or any alarm relay or other aftermarket installation. The fix for this is very simple: either find and replace the weak component or install a Ford starter solenoid. Yes, you need Ford parts to make a Chevy run, ironic, huh?
Anyway, let us know what you find and once we know what it is, we can walk you through the fix easily.