Don't think about speed and such - there is a much easier way to understand relativity. From energy.

It should be obvious that if you travel fast you have energy. And the faster you go, the more energy, and the heavier you are the more energy.

Now comes the interesting part: That velocity energy you have increases your mass! As soon as you have this, some very interesting things happen.

As you get faster you get heavier, the heavier you are the more energy you need to go even faster. This positive feedback loop eventually results in you needing an infinite amount of energy to go any faster (since you are so heavy).

The point where you need infinite energy is at the speed of light.

Now comes the second interesting effect.

Suppose you accelerate to half the speed of light - you have a certain amount of energy - no problem.

Now someone else does that too - only in the opposite direction.

He too had no trouble accelerating, and needed only a finite amount of energy.

But what happens if the two of you are traveling toward each other? A problem - the total of your speed is greater than the speed of light! And even if it was slightly less, suddenly you got energy for free! (Because of the feedback loop going from 0 to 1/2 c takes MUCH MUCH less energy that from 1/2 c to 8/9 c. So where did that energy come from?)

That obviously is impossible - you can't get energy for free. Instead the very definition of speed is changed! You have no choice - you have to change it. The definition is changed in such a way that the energy remains constant. And if the definition of time and length has to change to make it work, so be it. Energy is more important.

Of course all this rests on a foundation: That the increase in speed results in an increase in mass. But this too is required. We already know that's it's possible to convert energy to mass. So we could use a trick - convert mass to energy, speed it up, then convert it back to mass and suddenly we have mass moving very fast without putting in energy. Nope, can't do that. What actually happens is that energy has mass too, and this doesn't work. (If energy didn't have mass you could do other tricks, like send the energy up to space, convert it back to mass and let it fall under gravity.)

Hopefully this should show you why relativity is required if you have conservation of energy.