MISahnov's solution is spot on, but I have an easier method for this achievement.
This achievement is for warping 75 times, and I'll clarify by saying that warping is classified as having a ship with a warp drive, spending one warp cell, and then warping to another star system. It does NOT count if you use a portal gate either on a space station or on a planet.
Many people will likely get this achievement naturally by following the various questlines, or just by trying to get to the center the hard way.
However you can make your life a hell of a lot easier by only warping to a different system once you have your own freighter.
For those that don't know, freighters are giant ships that operate as mobile bases, great for storage and they even have their own special mission mechanics.
You will have a scripted space battle around your fourth of fifth time warping. You will be asked to take down a few pirates attacking a freighter, and after you destroy all of the pirates, the freighter's captain will invite you aboard and will offer you the freighter for free. You can choose to reload your game if you're not satisfied with the freighter you receive if you want an A or S class, but it has no effect on the achievement.
Once you have your freighter, you will be given a tour of sorts and then be given free reign to do as you wish. What makes freighters special is that 1 warp cell equals five warps when warping through the freighter controls. So instead of tracking down thousands of resources and wasting many spaces in your inventories, you can instead make 15 warp cells, and just sit in your ship and warp around however many times you need!
Hope this helps! Let me know what I can do to improve my solution!
Edit: According to the comments, it looks like the Origins update removed the 5:1 for warp drives. Sorry everyone!