My 2003 chevy will change to second gear but won't change again to another gear

I have seen where the power supply to the solenoids is failed. Check the correct wire to verify power at the transmission. The computer will indicate that it is telling the transmission which gear to be in, but the solenoids have no power and cannot actually turn on. This is limp-in mode. It is definitely a power supply issue to the trans solenoids.

Well code 1031 is a problem with the O2 sensor circuit which would not keep the engine from starting.
Code 731 is a problem with first gear and 732 is a problem with second gear, and again neither code should keep the engine from starting.
You may have a computer problem, or something going on with one of the wiring harnesses.
Have you figured out if you are lacking spark or fuel to keep it from starting ?

problem, with speed sensor

Sounds like the shift cable has broken

PRNDL Display
The instrument panel cluster (IPC) displays the selected gear position as determined by the powertrain control module (PCM). The IPC receives a class 2 message from the PCM indicating the gear position. The PRNDL display blanks if:
• The PCM detects a malfunction in the transmission range switch circuit.
• The IPC receives a class 2 message indicating the park position and the column park switch indicates a position other than park, open input.
• The IPC detects a loss of class 2 communications with the PCM.

Could be the transmission range sensor . Testing would need to be done , hooking up a scan tool ! Doing electrical testing etc...

Changing fluid is just that

Old fluid out, new in, won't fix a thing

Try cleaning the MAF sensor. Computer uses that info to determine pump pressure. If that doesn't work, you're looking at a new tranny

You might have burned out the transmission plates, try getting the transmission out of the car and inspect the plates inside.

no gears indicates that the torque converter clutch servo/solenoid s faulty
probably a disconnected wire from your working with the others

The alternator is not charging,or the battery wont take a charge.If all accessories work fine when its running after you've boosted it,its likely the battery.
John