You can reach the edge of the galaxy without traveling very far. My first time was quite a short trip, only 1,400LY or so straight "up" above SOL. Took just a short while in an Asp Explorer, pretty easy because the galaxy is thinner in the vertical. The farthest point I've done this was on the opposite side of the Milky Way galaxy, about 65,000+ LY from SOL, which took a few months of travel.
If you go on a trip with intentions to explore the galactic fringe, consider stocking up on the materials required for a 100% FSD boost injection. I used maybe 5 or 8 of these on my farthest trip, right near the edge. When a star was too far for my Asp, I could often use an FSD boost to reach farther (around 66 or 67LY max). Now with Engineer upgrades PLUS the FSD boost injections... You can get jumps even farther.
Carrying an SRV does hurt jump range a bit, but I found many, many landable planets and resupplied my FSD boost materials supply while out there. Back then it was trial and error surface prospecting, but today the Detailed Surface Scanner can tell you if what you need is on a planet.
Those elements are going to be key to getting as far out on the fringe as possible with current technology.
But I must caution: It is possible to use a series of FSD boost injections going out, and potentially not have sufficient materials resources to retrace those steps back. This would mean being stuck forever, unable to get back (well, outside self destruction!). While I was out on the fringe I made careful written lists of each time I used an FSD injection, the size of the injection, where I jumped "from" and "to", and insured that I always had sufficient materials stock to retrace that exact route backward.
Fly safe, and happy exploring!