It depends how far away those galaxies are, but yeah. The universe is expanding at an increasing rate. So, we have a light horizon beyond which we can't send or receive any particles, if they were to be sent right now. >>7572669
Has a good description of a constantly expanding universe.
Because we're in an increasingly expanding universe, our light horizon *used to be* farther away than it is today. The photons that started their journey near our light horizon in the distant past had to fight against less expansion. So, we can receive photons from further away than our present light horizon. So, the "visible universe" (the sphere from which we can receive photons) is bigger than our light horizon. >>7596369
You're describing special relativity, which is *relevant* to space travel but not the effect you care about the most. SR time dilation is a matter of perspective. The person in the space ship and the person on the planet both observe the other's clock to run slower. Hence the whole "twin paradox" thing.
What's *not* a matter of perspective is the acceleration the person in the rocket feels. That acceleration introduces additional time dilation. So, if you hop in a space ship and fly around, you age faster or slower than someone on the ground, depending on how much acceleration you experience relative to the surface gravity of someone on the ground.
If you wanted to skip 1000 years into the future, you'd need to *accelerate* faster than 1g for some time.