Fast Car

Ever since I was first told in a science class that nobody can travel faster then the speed of light, I've pretty much believed that claim. It would seem to be a reasonable theory.
Just yesterday, though, it dawned on me that this theory has a little glitch in it, or at least to my simple mind it seems to be a glitch. Someone straighten me out here if this idea is based on false understandings of science and Einstein's theories.
The thing is, Einstein theorized in such a way that if you were to put a clock in a moving object, let's say a spacecraft, the clock would be observed to run slower the faster that spacecraft moved. I think scientists have been skeptical about this but it is my understanding that this has actually been tested and indeed the clock ran slow as predicted.
Assuming that is the case, let's enter into the imaginary...
Imagine that mankind discovered a way to accelerate a spacecraft to somewhere near the speed of light which happens to be 186,000 miles per second, or thereabouts.
Imagine such a spacecraft existed and it had a clock in it. Not only that but imagine it had room for you inside of it as well.
Now imagine you were riding in this spacecraft at close to the speed of light. Got it in your mind? Good! According to Einstein, your clock would be running slow. I don't know the math but let's imagine for argument sake that at 180,000 miles per second the clock is running at only a tenth of its stationary rate.
Now imagine you flew right past someone else up in space who was stationary, not moving much. Let's call that person the stationary observer. The stationary observer also has a clock that she is watching.
From the stationary observer's perspective, one second after you pass by her, you will be a distance of 180,000 miles away from her. You will have traveled that far. In fact if you glanced out your window you would no doubt observe that you had traveled that far.
But...!!! Look at your clock! Since your clock is running slow, it was only one tenth of one second ago that you passed by the stationary observer. You traveled 180,000 miles in only one tenth of a second according to your own perception! - according to your own scientifically calibrated spaceship clock...
OK, so let's say from the stationary observer's point of view, another nine seconds pass by. From her perspective you have traveled 180,000 miles times ten seconds since you did your flyby. You are now 1,800,000 miles away. But on your very own clock, even though you indeed are 1,800,000 miles from the stationary observer, only one second has passed by since you passed that person.
You are traveling at 1,800,000 miles per second by your own speedometer! - 10 times the speed of light!!
This effect would only become more and more pronounced the closer your spacecraft approached the speed of light as observed by stationary observers. Maybe you could slow your clock down enough to travel a billion miles per second according to your own spacecraft clock! Maybe you could go even faster than that!
Was Star Trek's "warp speed" not so far fetched after all?
Fix my faulty thinking, someone...