Sounds like your thermostat isn't opening.
When you start the car from cold, the thermostat is shut so as to not allow coolant to circulate through the radiator. This helps to get the coolant to operating temp as quickly as possible. Once you reach operating temp, thermostat should open to allow flow through the radiator and provide cooling........this doesn't appear to be happening in your case.
Thermostats are inexpensive and replacement and testing is covered in the Haynes manual.
If you haven't got a Haynes.....let me know, I'll post some info for you.