You know that 運転 has to do with driving, right? And you know that 手 often has to do with a person doing something in words like these. So, a person who does driving is what? A driver.
The reading is all on'yomi, because it's jukugo. You should know the on'yomi reading of 手 by now just by seeing similar word patterns to this one.
When I took the taxi, the driver happened to be Tofugu's Koichi.
I ended up getting in an argument with the bus driver.
The driver seemed okay but his car was totaled in the accident.