High Water Bill - No Leaks
Make sure all the bills you look at are based on actual meter reads. Some utilities will estimate some of the bills (say, every other bill) based on historical data to avoid having to come check the meter as often. Since you haven't been in the house very long it's possible they're using historical data from last year (maybe the previous residents filled up a pool in the summer, or watered the lawn?).

If that's not the issue, see if you have access to the water meter. If so, write down the reading before you leave the house for the day and then again when you come home. 

Also note that a "typical" (whatever that means) person uses about 50 gallons/day, which for you should work out to about 6000 gallons per billing cycle. So these numbers don't seem way out of line.

EDIT: reading over these numbers now I can't help but notice that over your two billing cycles you are averaging 50 gallons/person/day, which is exactly the EPA's "typical" number. I wonder if your first bill was estimated and the second one was high to make up for it.

Also, I think the faucet thing is a red herring. If the faucet was leaking thousands of gallons you would notice it. And there's no way for someone to steal a meaningful amount of water from an outdoor spigot.