Electrical – Power Surge Damage to Refrigerator

Think of the electricity flowing into your house just like water flowing through a pipe. If you turn on a faucet only halfway then you will probably not notice a drop in pressure elsewhere in the house. But if your washing machine is filling (or your irrigation system kicks in, etc.) then you will see reduced pressure elsewhere in the house.

The electricity in your house works the same way. Air conditioners, refrigerators, water pumps, draw a lot of electricity when they first start up and you may see a drop in "pressure" when any of those items turn on:

Air conditioners, refrigerators, freezers, furnaces, etc. that have large motors require enormous amounts of power when the motor first starts to spin up to speed. Once the motor is spinning, the energy requirement drops way down. This instantaneous need for motor shaft speed is what causes that momentary dimming of your lights.
Reference: http://www.askthebuilder.com/QA_Lights_Dim_When_AC_Comes_On.shtml

As @Tester101 mentioned in the comments, you could look into a service upgrade to your house (which might not be a bad idea) but the problem might also be with your water pump. The pump kicking on/off every 15 seconds may indicate a problem with the pump itself (capacitor or pressure leak) or possibly an incorrectly sized pump for your usage. You probably want to look into a new/repair/upgrade of the water pump.

Are you sure your numbers are right? You say your kWh usage has about doubled, but your bill has increased 6x.

Anyway some utilities only do an actually meter read ever 2 or 3 months, then they estimate the bill for the other months. That can lead to extremely high bills if they've been underestimating and then they finally do an actual read and bill you for the extra that you've used. In theory the company is supposed to track your usage over time and get a good estimate but in my experience they can be wildly inaccurate. Your bill will tell you if the meter read is actual or estimated. Also note that some utilities increase rates in the summer when demand is high, so you may be suffering more if they just switched over to the summer rates.

If the problem is not just odd billing practices, you will need to dig a little deeper.

To help you narrow things down a little more, I would try to keep a log of your meter. E.g. try to write it down every day, or even twice a day to find out if your electric usage is constantly high or is intermittent. (Note that reading an old-fashioned dial meter is a little counter-intuitive, since each dial spins in opposite directions. See How to Read Your Electric Meter).

If you're really using 1000+ kWh extra each month, that's an average extra draw of 1.4 kW, which is pretty large. You should be able to see the meter spinning very quickly. Try shutting off circuits one by one to see what will slow the meter.