they just randomly show up while you are playing

Elu wrote How can I get Public Events to happen so I can do them?

Apparently, they just happen when you're patrolling the cosmodrome, on any planet i'm guessing.

Some examples are:

Defeat the Extraction Crews at three different zones.
Defend the Warsat from waves of enemies.
Destroy the Fallen Iron Walker/Devil Walker.
Kill the Enemy Target before he reaches his destination.
Prevent Vex sacrifices by killing Vex before they reach a sacrifice altar.
(From Destiny Wiki)

Also, i'm not sure where but on earth the Spider tank thing from the first Strike (the devils one) is suppost to spawn and it's considered a public event maybe you should patrol the Earth for a bit, and apparenly every 25-30 minutes it will spawn.