When chasing someone, you want to go W-E-W, rather than W-W-E. This is because his mimiced shadow slash slows (by quite a lot), but Zed's own shadow slash doesn't.
When ulting somebody, if you deal enough damage for the deathmark to kill them, a shuriken will appear above their head. Beware of being baited by Heal/Zhonya's, however.
Generally, when all-inning someone, you want to R-W where you think the'll run-E-Q-BotRK-Auto, and then continue to orbwalk after them, swapping with your W if necessary until you've secured the kill and/or are at risk of dying, at which point you'll want to swap back to your R.
In most matchups, you'll want to start either Longsword-3 Pots or Doran's Shield, depending on how aggressively you want to play pre-6. You'll generally want to pick up BotRK and Ghostblade as your first two items, after which you can build situationally (LW, Hexdrinker, Merc Scimitar, etc).