These tips include ways that I manage happiness on Emperor level, but they should work on Prince.
1) In the early game you should give potential city sites with new (to your civ) luxuries a higher priority. The same goes for potential conquests.
2) Puppet captured cities if you intend to keep them, you can always annex them later when your happiness situation is more stable. Cities you've just captured are in "resistance" and useless for micromanagement anyway.
3) If you are playing wide (5+ settled or annexed cities), take Meritocracy from the Liberty tree and connect all your cities with trade routes. If you are playing tall, take Tradition and get the Monarchy policy.
4) Unless you have a specific reason to choose Honor, I would recommend taking Commerce or Rationalism as your second policy tree. Each of those trees contains one policy that will improve your happiness. Honor does have policies to improve happiness, but these are reliant on maintaining standing garrisons and building lots of defensive structures. If you are going a warmonger route, this may work well for you; if not, you might want to use a different policy tree.
5) Befriend Mercantile City States. You don't even need to ally them (although that helps because they always have luxuries). Just being Friendly with them gives extra happiness per turn.
6) If you have a wide empire and make the Industrial age, taking the first policy of Order as soon as possible is always a good idea. 1 global happiness per city really adds up if you have 20 cities, and the bonus applies to puppets too.
7) Build circuses wherever you can and whenever appropriate. +2 happiness for no maintenance fee is more helpful than it looks. Also build stoneworks wherever you can, as appropriate.
8) Choose beliefs such as Ascetism, Ceremonial Burial, or Pagodas that can be used to generate additional happiness for your empire. In fact, Pagodas are an awesome, maintenance-free way to increase happiness while also boosting faith. Some Pantheons, such as Goddess of Love can be helpful in generating happiness as well (although GoL is better for empires where most cities can be expected to reach 6 population).
9) If you are going to annex a large number of captured cities, you may want to consider Autocracy to reduce the cost of building/buying courthouses. I'm not sure the benefits outweigh those of Order, though.
I hope something in this list is helpful to you. I find it helps to think of "Happiness" as "Stability", and to try not to do things that might destabilize your empire (e.g. annex 10 cities at once).