The decision to switch or not is really a decision between door 1 and door 2. At this point isn't your first guess completely irrelevant?
I've heard this problem and the reasoning before, and for some reason I can't get myself to believe that switching is better.
Assume you stick. In that case, you have a 33% chance you've picked the right door, as before. Opening another door doesn't change whether you've picked the right door or not originally.
Assume you switch. In the case you've picked a door with a goat originally (67%), you will necessarily switch to the door with the prize. In the case you've picked a door with the car originally (33%), you will necessarily switch to a door with a goat. Therefore switching gives you a 67% chance of picking the door with the prize.
Another interesting strategy to consider is "randomize" - what if you picked randomly between the two remaining doors? Then you'd have a 50% of getting the prize, since you're selecting randomly between two doors, one of which you know contains the prize.
For doubters: Before the door is opened by the host, you are less likely to have chosen the right one(with prob 1/3). Once he opens the door, he eliminates one "bad door". Now, given that you were less likely to have chosen the right one in the first place and one wrong one has been removed, it is highly likely that the residual door is the right one.