That you can’t go faster than the speed of light is a direct, simple consequence of everybody agreeing on the speed of light. It's a real-life Zeno paradox.

Alice and Bob are in some spaceships by a long racetrack in space. Alice fires a very brief pulse of light down the racetrack, maybe we see the rays that don't go straight through successively illuminate some rings around the track. Alice challenges Bob to race another light pulse, Bob revs his engine.

As the countdown hits zero, Bob accelerates to speed c/2 relative to Alice, then checks the reflected light from these rings only to find out that the light pulse is still traveling at speed c away from him. So he drops a beacon at his current speed then accelerates to speed c/2 relative to that, but no dude: the light is still moving at c away from him. Bob realizes that he can never win, so tries to instead measure the speed of Alice, who he expects to be moving at speed c away from him, only to find that she is instead moving at speed 0.8 c away from him.

Now the question is, how can this be? Consider a much slower spaceship. When Alice fires the light pulse and Bob starts moving forward at a slow speed, Alice sees this bubble of light expanding in kind of a uniform sphere centered on her. (Say it reflects off of space dust instead of a track.) Since they were at the same position when the light Bob also sees an expanding bubble of light, with himself at the center. The weird stuff about time dilation and length contraction does not apply at low speeds, if Bob goes at c/1,000 say, then these are only one part per million.

They both basically agree on how far this bubble of light is from Bob, in the directions perpendicular to the motion. The motion is parallel to the bubble in those directions, and to first order those parallel lines will not get any closer or further away. (This is why I want Bob to move at a slow speed!) They only disagree along the motion. Alice thinks the light is receding from Bob at speed c–v ahead of him, at speed c+v behind him: Bob sees the light recede at speed c in both directions.

So they come back together to repeat the experiment and Alice decides to force the contradiction. Alice puts a clock ticking out every millisecond out at distance 1 light-second, but it will start at the moment she fires the pulse, stopping when the pulse hits it, showing 1000. She puts one of these in the direction Bob will travel, and in the opposite direction. Surely he must agree that the light started from here and that it intersected those two clocks when they both said 1,000.

Bob agrees that the clocks look synced up and films all of this with a high-speed camera to make sure that there is no funny business, and they repeat the experiment.

Right when the light hits, Bob accelerates at his usual 1,000,000 gees for 0.03 s, to get his final speed of c/1,000. (I need to rewrite this to make the numbers more reasonable LOL.)

Here's where something weird happens, and it is entirely contained to those first 30 ticks of both cameras, as Bob looks at them in his high-speed camera footage. Bob corrects for Doppler shift like you do, and agrees that these clocks appear to be ticking during the other 970 ms at one tick per ms, his camera has maybe microsecond resolution and not the nanosecond resolution you need to see time dilation.

But during those first 30 milliseconds when Bob was accelerating, even after correcting for the Doppler shift, his best guess is that Alice artificially slowed down the clock behind him and artificially sped up the clock ahead of him. Because the clock ahead of him definitely ticked 31 times in those 30 ms, while the one behind definitely only ticked 29 times. So Bob says the light did hit these clocks when they said 1000 ms, but the clock ahead of him should have said 999ms at that time, and the light should have gone past it a bit by 1000ms, while the clock behind should have said 1001ms, and the light was actually not yet there at 1000 ms.

This anomalous Doppler shift is proportional to both the distance of the clock you're looking at, and your acceleration. It is also called the relativity of simultaneity, and it is the only new prediction of relativity, in that length contraction and time dilation are second-order consequences of it.