How best can I discover what is up with my electrical bill?
For devices that plug into an electrical outlet, you can use Kill-a-Watt or equivalent to monitor how much electricity they're using.

If that doesn't give you a clear culprit, one low-tech method is to look at the electric meter as you turn off circuits at the service panel one at a time until you see a big change in speed at which the wheel is spinning (for older meters) or the digits are changing (newer meters).  If you can narrow down what's using a lot of electricity, you can look at devices one at a time on that circuit to see if one is using more than its fair share.