I was about 1,000 light years from the start and jumping from star to star, scooping, advance discovery, then checking the system map. It will say "discovered by John Smith" over each planet or it won't say discovered by anyone. Fly over to it and do a detailed surface scan, then take the cartographic information back home and sell it. It will say you were the first to discover them.
This is pretty much my Elite career in the game. If you want to explore this is basically the routine:
a.) get a good exploration ship (Diamondback Explorer, Cobra, ASP, etc.)
b.) buy/equip the right modules (highest class FS drive you can equip, advanced discovery scanner, detailed surface scanner, fuel scoop, a repair unit, heat sinks, etc.) get rid of your weapons, you won't need them and weigh you down
c.) put the advanced discovery scanner in your fire groups
d.) plot a course within 1,000 light years (that's the galaxy map limit) from wherever in the bubble you are now. It also helps if you plot a course using only scoopable stars (KGB-FOAM class stars) so that you never have to worry about ending up in a stretch of T-tauri or brown dwarfs that you can't get fuel from. Continue adding 1,000 ly increments as you reach your destination to go deeper into the black for as long as you want to go out or towards something you want to see (Sagittarius A, nebulae, etc.)
e.) jump from system to system using your advanced scanner. I use the B-right/B-left shortcuts to quickly bring up the system map. From looking at the system map, I will typically know instantly if there's then anything I want to scan further.
f.) basically, if you hover the reticle over the star from the system map you'll have an immediate idea if they've been discovered or not. It'll say "Discovered by [Gamertag]" underneath the name. More than likely any planets in that system will have been discovered too if you see that (although that's not always the case)
g.) you probably won't find many undiscovered systems for the first 20-50 jumps after leaving the inhabited space. This differs though for everyone, some trips I've taken I found undiscovered stuff within a handful of jumps, other trips I was basically running into everyone's discoveries for 50+ jumps.
h.) when you find systems that are unexplored, scan the star, then look at the planets from the system map. There are clues to tell you if they are worth exploring with your detailed scanner or not. Plain white worlds are likely all icy bodies and worth extremely little (vs. the time investment) to scan them. But high metal content worlds, water worlds, Earth-like worlds, and most gas giants (with or without rings) are all generally worth your time and effort. You can actually even listen to the sounds the planets make on the map if your audio settings are right, they will give you clues/hints as to what they are.
i.) some things are worth scanning even if they are discovered. Things like earth-like worlds, water worlds, neutron stars, white dwarfs, and black holes (I've yet to find a black hole that was undiscovered already). However, though you won't get the "1st discovery" bonus or your name on them, they will still net you a good amount of credits when you bank them.
j.) rinse, repeat until you're just plain tired of it or delirious from the trip
k.) go home or to one of the systems out in the black (there are systems along the Colonia Highway) that you can bank your scanned data. Whatever you turn in before anyone else does that was unexplored at the time of scanning will become credited to you. I banked over 25 million credits on my last trip, and probably have my name on over 500 systems at this point.
