Help please!  I have a 80 vette that I've only had a couple months. The original engine(305) was replaced with a 350 not long before I purchased it(so I was told). I believe it has a 350 turbo 3 speed auto transmission. Today when I was driving home from work the transmission wouldn't shift out of 1st gear. I even tried shifting it manually but still wouldn't change gears. It will shift onto reverse. The fluid is full.

Good Afternoon,I too have an '80 Vette and like yours was bought and sold in CA.,i .e. 1980 Corvettes sold in Ca. because the 350 c.i. engines did not get certified in CA. only the 305 engine was available,however that has nothing to do with the "no shifting" problem.

In addition to what Sal suggested, there are only two other things that could keep a 350 from up-shifting...the governor, or a stuck valve in the valve body. Stuck valves in a 350 are RARE, and usually anytime a valve sticks in the valve body it is due to shrapnel from something else in the trans taking a doo-doo(yes...that's the technical term for explosions).
Check your vacuum line first. Make sure it is connected on the top, AND on the modulator valve on the trans itself. The modulator is on the passenger side rear of the trans. With the line properly connected on the engine, with the engine running, you should have vacuum AT the modulator line. If you do not, the metal line has rusted thru somewhere, or it is kinked. Even with NO vacuum to the modulator, the trans WILL still shift if you wind it out, and shift it manually. If it still doesn't shift like that, then the governor is prolly bad.
If the vacuum system checks out, then it's time to check the governor, located behind the large round cover on the driver side rear of the trans. There's a metal clip that holds the cover secure that will need to be removed first. With the clip off, you'll need a long flat-bladed screwdriver and a small hammer. Put the screwdriver against the flat edge of the governor cover, and tap it off with the hammer. It won't just fall off...you'll have to convince it to come off. With the cover off, the governor will twist out of the case. On the end of the governor opposite of the cover is a small plastic gear. That gear will either be sheared off, or worn in the middle(apple-cored). If damaged, the gear will need to be replaced, which is simply a matter of knocking out the roll pin that holds it into the governor, and removing the plastic that is left of the gear. Installing a new gear will require a hole to be drilled into the gear, as new gears do not have the hole for the roll pin. Insert the plastic gear fully into the governor, and drill the hole all the way thru, then insert the roll pin(the new gear will come with a new roll pin). Install everything in reverse. Now your trans should shift. If not, there is a stuck valve....which is not good.

Very good and concise Joel.As usual you explain these things in great detail.I have had to clean many governors in the past and was hoping he could get by with checking the vacuum line to the modulator.Have a great day.

Thanks for everyone's help. Turns out it was a bad governor😄

Glad it was only that!