If a star is 20 light-years away and I look at it through a telescope, what I'm observing is what the star was doing 20 years ago yes? So, if I fix my sight on the star and move toward it (for example's sake, let's say 0.5C or half the speed of light in a vacuum) in a linear path (which is not practical for real space travel) it would take me 40 years to reach it. By the time I reach it I would be observing in (almost) real-time what is happening. So to me that says that I've watched 60 years of the star's history in 40 years worth of time. So then, would it look to me as if I was watching a movie of the star's history in 1.5 times 'normal' speed? (60/40 = 3/2 or 1 1/2)
Yes, it will be 20 julian year ago.
Yes, it will take 40 years to reach there with speed half the speed of light through vacuum without any obstacles and star must be stationary(Though u cant travel with speed of light since physically its not possible to achieve that speed)
And as you are travelling with speed half to that of the speed of light you will experience a phenomenon of time dilation. Check all the details about time dilation here:
You will see time dilated by gamma = 1/sqrroot(1-v^2/c^2), factor so if you substitute this values it will take you 60 years to reach that star. and it squares all the things.
Hope this explanation helps you. If it is not understandable do let me know i will explain it in more mathematical way.