[@TheUnknowable]
Mainly major landmarks. In Asphodel that might be as simple as "A Tree", if it is as barren a wasteland as you describe it. Elysium's garbage dump might be another major landmark in Asphodel as people try to find something worthwhile in the castoff.If spirits feel the need to eat even without a body, is there a market of some sort? ("I got an orange rind that still has the citrus scent!") There might be a few buildings and monuments from times when restless spirits tried to find a way to occupy themselves. 

On the Tartarus side, go full Dante's Inferno. Describe some of the other places of ironic punishment. Hell (rim-shot), have fun with some of the cultural dissonance between the Greeks and modern society in terms of what they consider a punishable offense. Is there a processing center or courthouse of sorts? What features are used to keep the damned souls in Tartarus? Is it possible for people from Elysium and Asphodel to visit, if they were so inclined? ("Damnit I want to feel something again!")

For Elysium, what form does "Paradise" take? Pastoral fields meadows where the weather is always pleasant and the game plentiful? A never-ending carnival? Lavish ballrooms evoking a bygone era of elegance? A thumping nightclub where the more hedonistic desires can be indulged in without restraint or hangovers? Is the Greek Equivalent of Saint Peter literally a bouncer to this club? ("Sorry but you're not on the list")