If the engine will not crank over, the most common problem is the starter. These will tend to go out at about 100,00-125,000 miles. Sometimes it is only the starter solenoid, but often the complete starter (including solenoid) needs to be replaced.Google+
Car Problem Reports
Geo Prizm Vehicle Won't Crank Due To Starter
Geo Prizm Problem
Average Mileage: 206,500 mi (145,000 mi - 275,000 mi)
Visitor, 1990 Geo Prizm, 275,000 mi
Car does not like to start in cool damp weather the last few years after about 260-275K. Once warmed up and/or dry no problem. Starter clicks but does not turn - sometimes just hesitates a second sometimes several tries of 5-10 sec holding key to start position. strong batter helps but did not solve. Previously checked out but it always worked if in garage.
Gets worse every winter. Finally having starter replaced at 290000.
Had been trying to avoid too many more repairs after a major one earlier this year. Need to keep car drivable for at least another year.
Visitor, 1990 Geo Prizm, 200,000 mi
Just replaced the old starter. New one works great.