The rate that time is progressing is also going to change this effect a LOT.
Years are passing in seconds in this websites simulation, if you were experiencing it all in "real time" everything would appear pretty much stationary for quite a while.
Even at like 95% the speed of light, you're talking about years of time required to move between stars - nothing will appear to change much day to day until you actually get close to one of them.
You won’t feel the acceleration ;) nor the light intensity of the higher velocities (luckily).