You're on the right track to comprehending relativity, but there is still one piece of the puzzle you don't see.
The relativity you impose on time (by exemplifying that each observer perceives his own passage through time as 'normal') is also applicable to the three classical, geometrical dimensions.
To help you see this, let me clarify the difference between "dimension" and "coordinate."
*** A dimension is a fundamental aspect of the "fabric" of reality; a characteristic that cannot be attributed to anything more rudimentary. On the other hand, a coordinate is merely a position within this dimension.
We are restricted to dimensions, and can *never* escape any of them. However, we *can* change our positions among the coordinates of some of these dimensions (the three we call "space"). And, although we are restricted to a "set" rate of passage through the coordinates of the fourth dimension (time), we *can* alter our relative passage through it.
Regardless of a person's velocity he will not see any change in his own measured passage through time coordinates. Obvious enough, but to us outside observers, this person will appear to slow down or "dilate" his own time passage. As weird as that may seem, it gets even weirder - our traveler will also not be able to perceive any change in his own measured SPATIAL presence, though to us he will appear physically distorted.
Why is this so? Simple -- because time and space are inextricably interwoven, and anything that affects spatial characteristics MUST also affect temporal ones.
Remember, a dimension is a fundamental aspect of reality; a function that cannot be attributed to anything more basic.
Hence, you should be able to see that our imaginary traveler's rate of passage through time coordinates AND space coordinates will seem bizarrely altered. If time were not a dimension, only his spatial coordinates would be altered, and if this distortion is a function of a coordinate system, but not of dimensions themselves, then only the temporal distortions would occur.
There is one aspect about time dilation that one must mull over before he can fully appreciate what this theory implies. *Everything* traveling at high velocities is slowed down in a relative way - the subatomic processes within a person's fingernails, the rate at which the metallic vessel he's in ages, his digestion, his thought processes, you name it. Ergo, all activities we know as expressions of *spatial and temporal* are slowed, relative to outside observers.
Once you drop the Euclidean view, you'll see that terms like "speed of time" are as nonsensical as "length of length."