I just bought a 91 camry(5pd, 4 cyl.) The day I got it ran great started every time. Next night wouldn't start. I doesn't event turn over. I don't hear a click or anything. Friend advised battery-didnt work. I took to another friend who installed a new starter-still no joy. Right now I start with a wire he ran from the starter. I turn the key on and touch it to the + battery terminal and it fires right up.

Check the clutch start switch on the clutch pedal. You can disconnect it and run a jumper across the harness side. Try it then. If not, check the starter relay STA

That model also had serious problems with the ignition switch circuits. Many of the power feed connectors can be found to have poor contact, leading to connector housing getting too hot and melting.
I remember doing a bunch of Celica's too back in the day...