Temp gauge goes into red but, car doesn't run hot

Sounds like a little bit of air is still in the system. Bleed system like this. Remove cap from overflow tank fill with antifreeze to full mark leave cap off and start vehicle and let idle once vehicle starts to reach operating temp thermostat starts to open antifreeze level in tank will drop add more antifreeze to full mark then put cap back on. Also you will notice level in tank will start to rise before thermostat starts to open this is normal.

Ok. I'll try that. No need to buy temp sensor or new radiator cap then?

First answer is on right track, but you can't remove air from the overflow. In the top of the thermostat housing is an air bleed. You'll need to let the air out of this valve, and frankly play around with it until you get all the air out. System air locks easily, and it is difficult to get it completely full. Note, this is under the plastic cover on the drivers side where the upper hose connects. Also Note: This bleeder itself is frequently plugged up; you may need to remove it entirely to clean it out or just bleed it with the fitting removed entirely. If you here air bubbles go thru the heater core when accelerating from a corner, you don't have it all out yet.