What Can Cause A Chevy Transmission To Shift Hard Between 1st And 2nd?

 If it is a TH700R4, the shift timing is controlled by a throttle valve cable.
 If it is shifting hard, I would assume that it is also shifting later than it should. If you are applying the gas at an average acceleration, the first shift should come at about 20 MPH, The second shift should be at about 30 MPH and overdrive, (4th gear), should come in around 45 MPH.
If your transmission is shifting late, it will also shift hard. Maybe a throttle valve cable adjustment could cure the problem. If you look at where your accelerator linkage is attached under the hood, there will be at least one other cable next to it, (but maybe 3, if you have cruise control). The one that goes to your transmission, is the T.V. Cable.
If it's not a TH700R4, then it would be a 4L60E, which is electronically controlled. If that is the case, there could be a faulty shift solenoid.
Is your speedometer working? If not, it could be a Vehicle Speed Sensor. If you still have all the forward gears, a strong reverse and it is not slipping, I'm pretty sure that it will be an external repair. Which would be about $1500.00 less to fix than if it were internal.

You guys are all Geeks! Do you shoot your 10 year old dog when it gets sick????

Since you don't say whether it's a automatic or manual transmision I don't know...and since it is so old of a transmission there is no way to tell if someone has did any modifications to it like a shift kit,stall converter,I would say it is a automatic and there is low or no fluid in the transmission at one time or even the car/truck has been ran after the first signs of slipping burning up the clutch pack ...or the modulaor valve is out of adjustment...vacuum leak...this probably means the transmission needs to be gone through to find the problem...sorry