There are many, many, many unexplored systems. Everything that's been explored by all the players is barely enough to be counted as a fraction of a percent of the galaxy. The direct routes from the bubble to obvious nebulae are fairly well trodden but go up or down and you should find untagged bodies or systems within a few hundred ly. Once you get a few thousand ly out then actually seeing a tag becomes an event because it's so rare unless you're at an obvious tourist destination.
Check the Exploration Guides/Lists/Tools sticky at the top of this forum for a host of useful guides and info - and if you've still got questions after you've read that then ask away, someone will be sure to help you out.
If it's your first trip try something small - say a 15-20kly round trip to check you enjoy it. Going in the direction of Barnard's Loop gives you a cornucopia of brilliant views
and things to see or check out Veil Nebula West - probably my favourite nebula, looks like some kind of galactically scaled nougat treat. Both nebula names are searchable on the galmap.
Head into the black, send us pictures and most importantly have fun
