What stops my transmission from going into gear?

if its the the gear selector is moving and not going into gear ----under hood drivers side under brake mastercylinder is the top of transmission the gear selector cable connection is there check that it hasnt come off,loose or damaged ----=if selector does go into a gear and trans not responding look under drivers side and thru wheel wells for fluid leak ---theirs no dipstick so to check fluid
There is a black cap located on top of the trans almost directly under the brake master cylinder. It's about 2 inches high, 3 inches around. That's where you add the fluid. The checking procedure is as follows: Start the engine and allow the engine to idle until the transmission fluid temperature has reached the value specified. 104 degrees F.
Depress the brake pedal and move the shift lever through the gear ranges, pausing a few seconds in each range. Return the shift lever to the PARK range.
vehicle must be level with the engine running and the shift lever in the PARK range.
Remove the transmission oil level control plug.
The plug in an 11mm bolt located between the r/s axle seal and the oil pan.
Check the fluid level. The fluid level should be even with the bottom of the threaded plug hole.
If you remove the plug and nothing comes out, add about half a quart at a time until the fluid starts to dribble out. Once it stops, the level is good.this is the procedure to check fluid if their is fluid and problem still persists the transmission module and or internal transmission damage possible which a mechanic {transmission}would need to diagnostic check the following link has multi videos for the cobalt transmission 05 Cobalt Transmission Fill Yahoo Video Search Results

Transmission may now be toast since you continued to drive it when maybe low on fluid or torque converter had failed. How long since last fluid and filter change? Ever?

try to replace transmission strainer first torque convertor and see if that helps if not it probabaly the transmission

Do you feel any resistance in the gear selector when shifted through gears?

i think so as well ,think you may have cooked the gearbox computer

Check fluid level to see if low (car won't move with low fluid) Also is the gear selector feeling like its connected (no slop) if so could be loose and come out of gear, try moving it to other gears to see if it engages, if all these fail sounds like tranny has gone out.

Try unplugging the electrical connection on the transmission to lock it in 3rd. It might get you moving. Although the transmission is going to need rebuilt or replaced.

There is small chance you problem is with the slave cylinder, or the clutch master cylinder.
Check the brake fluid under the hood in the clutch master cylinder. If it is gone, or low, you may be leaking brake fluid either from the slave cylinder on the transmission, or the fluid could be bypassing the rubber seals inside the master cylinder.
I hope this is what is wrong, as it is much less expensive than having the transmission repaired.
God bless your efforts.

maybe you broke the main chain in the transmission // i've seen that happen on a 87 bonneville when starting off a green light with a full load of groceries in the trunk. they break just like anything else bad news is replacement is almost as cheap as fixing it / you need a replacement transmission :)