The runes are permanent unlocks that will give you access to locked parts of the map – including biomes. You will want to get them as soon as possible to not skip any possible loot on your way.
The Vine rune can be picked up in the Promenade of the Condemned by fighting with an elite Undead Archer (Tip: dodge its double arrow)
The Teleportation Rune in the Toxic Sewers unlocked by fighting with an elite Slasher (Tip: bring a shield or something to stun or immobilize him with)
The Belier’s Rune in Ossuary by fighting with an elite Slasher yet again.
The Spider’s Rune in Slumbering Sanctuary by fighting with an elite Orb Caster (Tip: bring a ranged weapon)
So if you’re in a hurry to get them all your routes would be:
The Prisoners’ Cells > The Promenade of the Condemned
The Prisoners’ Cells > The Toxic Sewers
The Prisoners’ Cells > The Promenade of the Condemned > Ossuary
The Prisoners’ Cells > The Toxic Sewers > The Old Sewers > The Insufferable Crypt > Slumbering Sanctuary
You can’t do them all in one swoop, so once you’ve unlocked one either carry on or die and start from the beginning. There’s also Challenger’s Rune that unlocks daily runs in the starting zone and can be found after defeating boss in The Black Bridge, but that one isn’t needed for progression and you’ll get it eventually.