If the car goes in the red again, shut it off and let it cool or risk serious damage. No idea if you damaged anything or not. If the car suddenly feels like it lost a bunch of power, than it's quite possible. Keep an eye on the coolant level. If it starts dropping but isn't leaking anywhere, check that the oil level isn't going up with coolant in it. (head gasket failure)
Let the car idle in your driveway with the hood up and see if the fan kicks on. If the car starts climbing past the halfway mark on the temp gauge and the fan never kicks on, shut it off. Then try and figure out if the fan is getting power or not. If NO power, look at fuses or the relay. If the fan gets power but doesn't turn on, expect a bad fan.
Your radiator might be clogged internally or externally. Internally means there are deposits built up in the cores and it will need replaced. If clogged externally, it means a bunch of dirt has seated in the fins and will need rinsed. Best way to rinse is remove the sprayer from a garden hose and hold the open end of flowing water right up the radiator. Watch the color of the water coming out the other side. Will be very dirty and cloudy water if clogged. Once water is clear, move to next section. This can result from driving down gravel or dirt roads too often as the dust gets kicked up and pulled into the fins, clogging over time. At idle, the airflow isn't enough and the dirt holds in the heat. Once moving, there is just enough air going thru to cool it again.
My bet is on a blown fuse or the relay if the fan isn't working. If the thermostat is sticking, it might be the problem, but usually it'd be more erratic with the temps, but who knows.