Suppose we play a racing game. I scatter a little bit of dust around space, then you come by me in your spaceship at some speed $v$. Let's start with $v = c/2$, just so we're not contentious. Right as you pass, I fire a really bright laser pulse in the direction you're going. You're racing the laser light. The dust means that you see reflections of it, so you can measure, in your coordinates, where you think it is.
Here's the basic problem: I (stationary relative to the dust) measure this light as moving away from me at speed $c$. You, moving relative to me at speed $v$, also see it moving away from you at speed $c$. The better your instruments, the better you will find that the light is moving away from you at speed $c$.
Now let's speed you up a bit more. By this point I am far away, so don't count on me to help you: instead, you drop a little marker in space and then accelerate until that marker is moving backwards at speed $c/2$ relative to you. How fast is the laser pulse moving away from you? Still at speed $c$. So you drop another marker and you accelerate to speed $c/2$ relative to that. Still, the light is moving away from you at speed $c$.
You cannot win. That is what the present theory says. Since I will always see the same events as you see, I will never see you outrun that light pulse. So from my perspective, nothing you can do short of magical teleportation will enable you to outrun the light pulse.
Let's say that you can measure the speed of me moving away from you -- or maybe you just measure the speed of the dust. It does not go at speed $(-1/2)c$, then at speed $(-1)c$, then at speed $(-3/2)c$ relative to your spaceship. That is not how velocity addition works in relativity. Rather, it goes at speeds $(-1/2)c,~(-4/5)c, (-13/14)c$. The dust never goes faster than light either.
You'll be heart-warmed to know that there are no paradoxes. We can prove that the mathematics is 100% consistent.
The reason that this all happens is that when you start moving relative to me, we start to disagree on the time that the "present" is at far-off places. These disagreements start to add up pretty quickly as you start to go a significant fraction of the speed of light, and so that when you get close to it we both see each other's clocks as moving slowly, and we both see each other's spaceships appear to become shorter in the direction they're traveling.
This gives you the answer to your second paragraph, too: because there is no "absolute frame of reference" that the speed of light is relative to (everyone sees light move at speed $c$), no, we can't determine our motion relative to that absolute frame of reference. (But, we know something a little similar: we have been able to determine our motion relative to the cosmic microwave background, which is actually pretty significant if you think about it; it basically says "we know how we're moving relative to our local part of the Big Bang.")