I was unable to avoid getting the highest tier of completition in any public event I was in (most of the time it's either shoot a specific target, or stay around a target). So I'm not sure what's required to get the highest tier outside of "just trying". Even joining late, or doing minimal damage gets the gold tier in my experience, dying doesn't appear to affect this (I died three times on my first event and still got a gold).
The highest tier is Gold, (Three gold "flags" will appear on the top of the screen when the public event is over)
Public events happen often, you'll get this over time if you play enough bounties and missions and just join any public event you can.
Bungie is currently setting it up so Public Events will happen every 15 minutes. When that hotfix goes through this will be a lot faster to complete.
You WILL see public events as you're walking towards your missions, strikes, and patrolling as you progress through the game. and again participating should give you a the highest tier of completition.