How does light on board a spaceship behave at (for example) 50% light speed To better explain my question than I can in a single sentence:
Case 1: I am (for example) on board a spaceship inside a large room with a laser in the ceiling and a point drawn on the floor at the location it hits while the ship is stationary.
Then after the ship reaches 50% light speed (Not accelerating) and i turn the laser back on will it still light up the same point?
Case 2 Inside the same space ship there is another room with a laser in the wall shining onto a receptor at the other side of the room. The receptor measures the time it takes for the laser to cross the room.
Then the same as before the ship is a 50% light speed and not accelerating. The laser is in the wall which is at the tail end of the ship and the receptor is at the front. I turn on the laser does it take longer for the laser to reach the receptor?
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These questions more or less come forth out of the idea that light always travels at the same speed. Further, when I hear people talking about the light outside the ship blue shifting but what about on board the ship itself? What happens?
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Edit 1: as a reaction to this comment and as an amendment

"Then after the ship reaches 50% light speed" - relative to whom? The spaceship is at rest with respect to itself, it's the other 'stuff' that's moving. – Alfred Centauri 9 mins ago

This comment seems to say that everything is relative but that seems strange because light cant go faster than well "The speed of light". But if this is relative then on that spaceship that (relative to the space around it) is moving at 50% light speed you fire a laser from the front of the ship will it move (relative to the space around it) at 150% light speed? That seems to be wrong? please tell me if something is wrong with my logic.
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Edit 2:
Okay i feel stupid now, i think i found the answer and it has to do with time dilation. I have heard that the closer you come to the speed of light the slower "your time" becomes. Which means that the answer to case 1 is: it hist the same spot, and the answer to case 2 is: from your perspective it still takes the same amount of time but if someone outside the spaceship could see the start and finish time it would take a lot longer.
What i have written above is what i think is the answer but with my rudimentary knowledge on the subject it would be nice if someone could tel me if i am correct and probably correctly rephrase it.
 A: 
These questions more or less come forth out of the idea that light always travels at the same speed. Further, when I hear people talking about the light outside the ship blue shifting but what about on board the ship itself? What happens?

Nothing that you can detect, it will all seem normal to you inside the ship, even at c/2, the outside view will be much the same

Case 1: I am (for example) on board a spaceship inside a large room with a laser in the ceiling and a point drawn on the floor at the location it hits while the ship is stationary. Then after the ship reaches 50% light speed (Not accelerating) and i turn the laser back on will it still light up the same point?

Yes

Case 2 Inside the same space ship there is another room with a laser in the wall shining onto a receptor at the other side of the room. The receptor measures the time it takes for the laser to cross the room. Then the same as before the ship is a 50% light speed and not accelerating. The laser is in the wall which is at the tail end of the ship and the receptor is at the front. I turn on the laser does it take longer for the laser to reach the receptor?

To you everything looks the same, your pov is as good as anyone else's. You won't notice anything different from Earth. 
A: Case 1: Yes, the laser beam will still hit the same point.
Case2: The time for the light to reach the receptor will be exactly the same as before.
There will be no frequency shifts in the light inside the ship.
Additional note. Both cases 1 and 2 are a basic tenet of Special Relativity, i.e., that all physical experiments in an inertial reference frame have the same outcome, independent of the relative velocity to other inertial frames.
According to the velocity addition theorem of SR you will never be able to achieve a light speed larger (150%) than c. 
