The "Bubble proper" is a roughly egg-shaped region of space centred on Sol. It extends roughly 200 LY in the North, South, East and West directions ("North" is towards the Galactic Core), about 250 LY in the "Up" direction (towards alliance space) and almost 300 LY in the "Down" direction (towards Imperial space). The easiest way to see the actual boundary of the Bubble is to go to the galaxy map, select one of the "inhabited system" indicators (I normally choose "Allegiance", but you can also choose "Economy" or a couple of others), then make sure every option except "None" is selected. This way, all the inhabited systems are highlighted, and all the uninhabited systems are invisible. Then simply scroll the map around.
The Bubble has a surprisingly sharp "edge" - in most directions, the inhabited systems suddenly stop, rather than gradually peter out. It contains just over 20,000 inhabited systems, and roughly twice that number of uninhabited systems.
The method I outlined above doesn't do very well at viewing the "entire Bubble" all at once, as the viewing radius of the galaxy map in these modes is too small to include the entire Bubble. To workarounds for this: go to the Powerplay tab in the galaxy map, and all the powers are highlighted. This won't show the "entire Bubble", because Powerplay tends to focus on the high-populated systems,and the fringe of the Bubble is generally made up of low-population systems. So when you look at the Powerplay mini-bubble, imagine a further shell of systems surrounding everything, about 25 to 50 LY thick.
Your second option at imagining the Bubble is to bookmark the following systems, that are roughly at the six "poles" of the Bubble:
North pole: Pand (-10,-3, +208)
South pole: Maruda (-10, -14, -185)
East pole: Sabi (+174, 32, 23)
West Pole: Mechelkanu (-211, 21, -11)
Up pole: Gliese 460 (+17, +208, -17)
Down pole: Kilique (+90 -265 +7)
Once you've bookmarked those systems, zoom out, then imagine an egg-shape (with the fat end at the bottom), that sits inside all six of these points. That's the Bubble.
The vast majority of the galaxy's 6.6 trillion humans live within the Bubble, but there are a few outlying colonies outside the Bubble proper. There's the Sothis Spur, a string of colonies connecting the Sothis system (about 500 LY away) with the Bubble. There's the Pleiades colonies where all the Thargoid action is at, plus a string of bases connecting them to the Bubble. There are a few isolated inhabited worlds, like Pratchett's Disc in HIP 74290.
Finally, there are the deep space outposts, well outside the Bubble. Most of the "named" nebulae within 2000 LY of Sol have at least one outpost within it or nearby. There's a string of starports along the road to Colonia, and Colonia proper, a mini-bubble 22,000 LY away. Finally, there's a starport adjacent to the supermassive black hole at the galactic core, 26,000 LY away.
As far as "leaving the Bubble for safe mining", you want to travel at least about 500 LY from Sol - or about 300 LY beyond the edge of the Bubble in most directions. I think you can be near one of the remote outposts (like the Pleaides or Sothis) and still benefit from the "too far from Sol" effect.