The first symptom of a bad transmission shift solenoid is having an engine light on. If the transmission control module detects anything wrong with the sensors, it will turn on the engine light to warn the driver and set a diagnostic code. Some cars also come with a transmission light.
Here are some codes related to a faulty transmission shift solenoid:
P0750 (Shift Solenoid A)
P0755 (Shift Solenoid B)
P0760 (Shift Solenoid C)
P0765 (Shift Solenoid D)
P0770 (Shift Solenoid E)
Ensure you can access your car's diagnostic codes by acquiring an OBD2 scanner here.