I'm stuck on the Lily Path - slave quest, how do I progress? I do not know what else to do
After you buy Lily's BDSM set, that is where the story for v0.06 of Terrene finishes, so you've reached the end for this update.
The story continues in v0.07, which is currently on my patreon.
I'll release v0.07 here on itch, once I finish v0.08, which I'll start working on soon.