Sandbox play is one of the greatest styles of D&D (or any other RPG) campaigns because it relies on self-governance, rather than the GM making all major decisions.
Create your own sandbox campaign using this outline along with my follow-up here...
A. Wandering monsters - create a random table for PCs traveling from place to place
B. Hazards - come up with some sticky situations the PCs might encounter in their travels.
C. NPCs - who will the PCs meet along the way?
II. Noteworthy NPCs
A. Flesh out prominent NPCs in the realm - King, Queen, other royalty and nobility.
B. Lower-class patrons, mentors, merchants, and faction leaders who the PCs may interact with on a regular basis.
C. Nemeses! The PCs will have enemies pitted against them and their causes, thwarting PC goals, and possibly trying to kill them.
III. New Details
A. More information about places the PCs already know, like the small dungeon you told them about in the 1st session.
B. Further reaches - what's over the mountain or beyond the barrier or below the surface?
C. Weird shit! Now that the PCs feel rather comfortable and "at home" with the campaign setting, it's time to throw in a few curve balls. What gives that giant floating crystal the right to mind-control everyone who steps foot inside the obsidian labyrinth?
That's it for now. I might come up with part 3 in a few days. If you have a suggestion, question, or comment, please post something below.
p.s. I made a video where I talk about this blog post.