The easiest remedy for unhappiness is using the "Turn on avoid growth" button in the city screen, this is particular usefull when your civilization has not yet matured enough to take advantage of things such as Monarchy which effectively negates all unhappieness with the right military pressence.
Representation helps a bit too but nothing like Monarchy.
Other ways to get happy faces is through resources such as gold, silver, gems, fur ect. You have to connect the resource with a road for it to show up in the city screen however.
The green mist, or unhealhtyness is something much harder to deal with, but also less of a problem(in most cases) for each unhealthyness present above your total health you only get -1 food, which is easy enough to live with. Untill you can build things such as Granary, Aqueducts, Grocer, Harbor ect. you will see alot of this.
However, unhealhtyness can be helped by placing your cities away from deserts and jungels(which you can remove ofcourse) they add -1 health and having forests inside your cities fat cross, forests add +1/2 health.
Most often health is the least concern and can only be eased out of by the mid/late game.
Unhappieness is the real concern and can be kept in check using the tools mentioned above, but you have to micromanage this in some way or another(Such as using slavery to whip away population to get below the happy cap).
in your city.
