Manual Transmission Won't Shift Out Of 1st Gear, Why?

Most of the problems with manual transmissions are in either the clutch, clutch linkage or the shift linkage so try and look at them first. Both could cause your problem.

Fixing the linkage is not too difficult.

Fixing the clutch linkage might require removing the transmission to get at the throw out bearing and associated parts.

Fixing the clutch will require removing the transmission.

And once you get the transmission out of the car a mechanic should be able to tell where the problem is. If it is the transmission now would be a good time to find a good used transmission (from a salvageyard) and put it in.