When I worked as a computer guy in a factory, I always knew to have plenty of spare parts around for shutdown periods. If you leave a computer on for months at a time, the most likely time for a failure is when you shut down. They just might not come back up.
That story actually doesn't have a lot of bearing on your personal decision, other than point out the the power cycling down and then back up is a time of stress for a computer. But, it's really very minor stress. I assume your house is a lot easier on computers than the dirt, heat, and constant vibrations of a factory floor.
If you have a high speed internet connection, I'd shut down when you're away from the computer - especially if you don't have a good firewall.
If you're worried about the few pennies in electric costs, go ahead and shut down.
If you don't shut down, make sure you at least have a screen saver. Displays still burn pretty quickly.
Definitely shut it off if there's a lightning storm in the area. Even if you have "good surge protection". As if. Hehehe.
Have fun.