So, how do you differentiate between time frames?

Think about it objectively for a moment. How would you compare differently moving clocks? You could transmit some kind of radio signal from one clock to the other, but in that case there's a symmetry: an observer at A always sees the clock at A ticking faster than the clock at B, but the observer at B always sees the clock at B ticking faster than the clock at A. In special relativity, the local clock is always the fastest clock.

This might seem paradoxical, but it works just fine, because the only way to actually directly compare the clocks is to put them in the same room and look at them, which means one of them must be accelerated into the reference frame of the other, and it's that acceleration that breaks the symmetry of special relativity.

Whence comes this so-called 'time travel'?

Nowhere. There's no theory of time travel that doesn't contradict one or more of the fundamental truths of our universe.

I've tried to imagine this process and all I can see happening is that, as you approach the speed of light, it becomes harder and harder for anything to move.

Not at all. If you were accelerating at a constant rate in a small spaceship with no windows, you wouldn't be able to tell in any way whether you were moving at one mile an hour or six hundred million, relative to some arbitrarily chosen, notionally stationary point. (You also wouldn't be able to tell that you're not just standing in a windowless room on the surface of the Earth.)

If every part of your body was going at that speed, it would 'seem' to stop in time

On in the reference frame of an observer who's notionally at rest. (Or, conversely, you're at rest and this other observer is rocketing past you at an absurd speed. It's the same thing.)

because you can't really move in any direction without an enormous (near infinite if not infinite) amount of energy

Few things confuse new students of relativity more than the notion of "relativistic mass." That's why it's not taught any more. The idea is not a physically meaningful one, and it violates the equivalence principle to boot. In your own reference frame, you observe yourself having no more momentum than you would under any other conditions.

The normal processes that happen in your body, right down to the subatomic level, would be 'suspended' (and you might even become compressed).

Again, only from the point of view of a differently moving observer.

You would simply move through the universe at near light speed until you were able to stop.

Yes, but all distance intervals parallel to your direction of motion would be contracted by virtue of your velocity relative to the things you're measuring. That's how it's possible for a fast-moving thing to get from some point A to some point B in less proper time — that is, the time as measured by a moving clock — than you would assume by just multiplying the speed of light by the stationary distance. In the reference frame of a notionally stationary observer, time for the moving observer is dilated. In the reference frame of the moving observer, time works as always, but the distance from A to B is contracted.

The concept of time as a uni-directional dimension never made much sense to me.

That's because time is fundamentally different from space. There are a lot of different technical ways to explain this, but the simplest is to point out that the metric of four-Euclidean space — what spacetime would be like if time were just like space — has the diagonal terms +1, +1, +1, +1, while the metric of Minkowski spacetime is -1, +1, +1, +1. The time dimension is different from the space dimensions.

why would you even call time a 'dimension' at all?

Because you need four coordinates to specify a point in spacetime, and the dimensionality of a manifold is equal to the number of coordinates you need to specify a point on it.

Personally, I think time is nothing more than a unit of measure and doesn't exist as a dimension.

It's absolutely fine for you to think that. It won't have any negative effect on you or your life if you hold that opinion. As long as you understand that you're not actually, technically correct.