If you have a space ship and accelerate constantly you can replicate gravity while also going quite fast. You will quickly reach relativistic speeds, and will need to take special relativity into account. Accelerating at a rate equal to Earth's gravity will allow you to reach anywhere in the 100,000 light year across galaxy in under 25 years (ship time). Keep in mind that if you want to arrive stopped, you'd have to travel half the distance and then turn around and spend the second half of the journey decelerating.

If you don't have a space ship capable of unlimited acceleration, the math works for more mundane speeds as well.

All the math came from this site.