Why doesn't time contract? I'm tutoring a Year 12 (high school) physics subject which requires me to understand special relativity, in particular, time dilation and length contraction. I have only studied 1 semester of 1st year uni physics, so bear with me for sounding ignorant. I've tried reading parts of Physics for Scientists and engineers but have more questions than answers.
I get that if you're traveling away from a clock at high speed, the clock will "appear" to slow down with respect to the person. After one tick, you've moved a few more million metres away and so the light has further to travel to reach you, which will take longer. A bit like looking up at the stars and thinking that is it happening now, but it's actually something that happened thousands of years ago.
What happens if you move toward the clock. Does time speed up?
So why is there only time dilation and not contraction? My only understanding I can lean on here is the doppler effect, but I have a suspicion that has nothing to do with it.
PS I have a degree in engineering, but I struggle to get my head around this stuff.
 A: Special relativity is notoriously easy to misunderstand, and I should know as I misunderstood it myself for a long time before I got it.
The Doppler effect does apply in SR, and can cause events to appear to increase or decrease in frequency, depending on whether you are moving towards or away from them, and that is caused by the fact that the distance light has to travel from the events to reach you is changing all the time when you are in motion. Time dilation has a quite separate cause.
If you want to develop a rock solid conceptual appreciation of SR there are a number of key principles you have to work at until you really understand them.
One is that all all motion is relative, so all the main effects of SR apply symmetrically. For example, you might have heard that one of the facts explained by SR is that muons travelling through the atmosphere at close to the speed of light have extended lifetimes because of time dilation. Well, since all motion is relative, you are moving at close to the speed of light relative to a passing muon, and in the frame of the muon it is you who is time dilated and length contracted.
Time dilation is perhaps the most widely misconstrued effect in SR, probably the result of the common saying 'moving clocks run slow'. You shouldn't think that a moving clock somehow stops measuring time properly. If a clock is time dilated relative to your frame of reference so that it measures five seconds, say, while ten seconds have passed in your frame, that doesn't mean the moving clock is measuring time at a reduced rate- what it means is that only five seconds have passed for the moving clock, and the moving clock has accurately measured it as such, ticking away at its usual rate.
Time dilation is a consequence of the relativity of simultaneity, as is length contraction, so I recommend you focus on understanding that first. If you are interested, I've added an illustration below of how time dilation comes about and how it arises symmetrically.
Imagine you are walking along a line of people each of whom has a watch identical to your own and ticking at the same rate, but, and this is important, each watch down the line has been set one minute ahead of the watch before it. As you pass each person and ask them the time, your own watch will seem to lose a minute each time- ie it will appear time dilated. That is not because your watch is actually ticking more slowly, but because each watch you pass is a minute ahead of the last one.
Now, imagine you are being followed in your walk by a line of friends, also with identical watches to yours, but again the watch of each friend in the line is set a minute ahead of the friend in front of them. From the perspective of any of the people in the stationary line, they see you pass, then they see each of your friends passing with a watch that is a minute further ahead, so they think their own watch is losing a minute each time (ie time dilated).
Because all the watches are out of synch, everybody in the stationary line thinks their watch is time dilated compared with the walkers's watches, and every walker thinks their watch is time dilated compared with the stationary watches even though all the watches are ticking at exactly the same rate.
In SR this happens because a plane of constant time in one inertial frame of reference is a sloping slice through time in a frame of reference moving relative to it, the slope being upwards in the direction of motion. So if you are moving through a frame, the clocks ahead of you in that frame are increasingly ahead of your clock, the discrepancy increasing with distance, just as with the watches held by the stationary line of people. So although your clock continues to tick at the rate of a second per second, it appears to lose time compared with the stationary clocks you pass because they are each set successively further ahead of each other in your frame of reference.
And finally, what causes time to dilate rather than contract is the fact that the plane of constant time in the stationary frame slopes upwards in your direction of travel. If it sloped downwards instead, then all the clocks ahead of you in the stationary frame would be gradually further behind each other, and as you passed them you would think your clock was gaining time and thus be time-contracted.
A: You are correct that it has nothing to do with the Doppler effect. It is assumed in SR that you've subtract the time light takes to reach you in your time measurement of an event. So when we say an event as co-ordinates $(5m, 3s)$, then 3s is the time at which the event actually happened in your frame of reference, and not the time at which you saw the event happen.
Always refer back to this formula when in doubt:
$$(x_1-x_2)^2-c^2(t_1-t_2)^2=invariant$$
The $c^2$ may look weird. If it does, you can choose length and time units that make $c=1$. This formula is called the "space-time interval". It is sort of a distance between any two events in spacetime. For any two events $A$ and $B$, its value remains unchanged in all frames of references.
The consequences? If you increase, the space-separation, $(x_1-x_2)$, between two events, the time separation $(t_1-t_2)$ must increase too. So this is what happens with clocks:
You have a clock sitting at rest in your frame, say, $2m$ away from you. At $t=1s$, the clock ticks once. At $t=2s$, the clock ticks again. So the co-ordinates of these ticking events in your frame are $(2m, 1s)$ and $(2m, 2s)$. The space separation between the events is 0m. There's only a time separation of 1s.
An observer, with respect to whom the clock is moving, observes the two ticks of the clock at two different points in space. Since the space separation between the two ticking events has increased, the time separation must increase too.
But then why wouldn't length dilate too? Take a ruler of length $1m$ at rest in your frame. Take two events with space-separation as well as time separation
Event A (0m, 1s): The left end of the ruler existing at 1s
Event B (1m, 2s): The right end of the ruler existing at 2s
There's a moving frame with respect to which these two events become simultaneous evens (see Lorentz transformations ). Since the time separation decreases ("time-contraction"), the length separation also does. Since, in this frame, these two events are simultaneous events of the "existence of the ruler", this frames takes the space-separation between these two events to define the "length of the ruler".
The asymmetry between time-dilation and length contraction arises because time-dilation is defined in terms of the time-interval between two events. If we also defined length contraction in terms of the length separation between two events, then there would be no asymmetry. Length would dilate the same way as time. But length contraction is defined based on the length of an object. The length of an extended object is a weird notion in relativity, as it depends on what parts of the object exist simultaneously in your reference frame.
A: 
After one tick, you've moved a few more million metres away and so the light has further to travel to reach you, which will take longer. A bit like looking up at the stars and thinking that is it happening now, but it's actually something that happened thousands of years ago.

What you are describing here has nothing to do with time dilation. What you are describing is the classical Doppler shift. (The relativistic Doppler shift is similar but also includes time dilation)
In relativity, the “big three” effects (time dilation, length contraction, and relativity of simultaneity) are what remain after the observer correctly accounts for the delay in the signal reception due to the finite speed of light. They are not optical illusions, they are what remains after eliminating the optical illusions.

What happens if you move toward the clock. Does time speed up?

If you move toward the clock at relativistic speeds then there is a relativistic Doppler shift. This includes a classical Doppler blue-shift plus time dilation which makes the signal a little less blueshifted than you would expect classically.
The time dilation effect does not depend on the direction, so it even occurs if the clock moves perpendicular to the receiver, thus it is sometimes called the transverse Doppler shift.

So why is there only time dilation and not contraction?

It is because of the symmetry. For the first postulate to hold, the principle of relativity, all inertially moving clocks must go slow relative to any inertial frame.
You can get time “contraction”, but only where the symmetry is broken. For example, in a gravitational field, both clocks agree that the clock down lower is slow and the clock up higher is fast. The gravitational field breaks the symmetry since both can tell which clock is the higher and which is the lower.
A: Instead of two observers, it is more clear to think of one observer travelling between two points in the same frame, and the clocks of that frames were previously synchronized. For example, a rocket between Earth and a (future) Mars base.
If the rocket has a really big velocity, the crew will see at arrival, that the time of travel $(t_M - t_E)$ for the basis clocks is greater compared with their own clock. This is time dilation.
Any communication from Earth will be in slow motion, and from Mars fast forward, but it is not time dilation, its an effect of the relative velocity.
A: On a general note, I recommend the educational game Velocity Raptor to help improve your conceptual understanding of special relativity, both for this specific point and many other aspects of it.
For this specific question:
If you are moving towards or away from a clock, or equivalently if a clock is moving towards or away from you, in your reference frame the clock is running slow. This is true regardless of whether the movement is towards, or away.
Where the distinction between towards vs away becomes relevant is when you accelerate towards or away from the clock. When you accelerate, you change which inertial reference frame you are in, and consequently change your definition of "now" for distant locations.
If a clock is far away and at your current "now" time shows 6am, accelerating towards the clock might change your definition of "now" such that the clock "now" shows 7am. Conversely, accelerating away from the clock might make the clock "now" show 5am. This effect scales with both the magnitude of your acceleration and the distance between you and the clock.
The magnitude of this scaling is such that, if you accelerate towards the clock to arbitrarily close to the speed of light, your definition of "now" at the clock's location will jump forward just enough to match your travel time to reach it, such that it being effectively frozen during the trip will produce a result on arrival consistent with calculations done in other frames.
