Temp gage says it's overheating while idle but there is coolant

Last weekend my daughter's radiator tank was empty and started smoking so I added coolant. Since then the car has set pretty much for a week. She drove it home this weekend about a hour and a half drive she said when she got to town and had to idle at red lights the temp gage went to hot. I checked the car this morning completly cool and the tank is still full. Any suggestions? I will check the fan to see if it's working since I haven't done that yet but didn't know if there is something else I need to look for.

Also, she says she noticed when she goes about 70mph her speedometer doesn't work right.

Any help and suggestions would be grealty appreciated!