Actually the easyist way in Vanilla where there are only a few happiness sources to get a large number of cities and keep them happy is to only build a few, but conquer a lot from the AI.
Tradition for Monarchy, whose pop unhappiness reduction in the capital will exceed a mere one citizen per city for any empire you can keep global happiness positive with, especially for Vanilla that doesn't have the religious sources of happiness.
Liberty is for if you really want to self found 6+ cities instead of conquering one due to the massive cost reduction of settlers. (It takes too long if you don't go Liberty to hand build 5+ settlers.)
Colosseums, Circuses, Stone Works, everywhere. Build Circus Maximus for a free source of 5 more happiness (requires every self built city you already have to have a Colosseum)
I forgot what they called Zoos back in Vanilla, but you'll need those everywhere in Vanilla as well, especially if going wide.
After your first tree, for both Vanilla and G&K there is actually within Rationalism a policy that gives happiness bonuses to science buildings which is another thing you should build everywhere.
If happiness is still a problem in late game (this is much more likely to be the case in Vanilla than in G&K or especially BNW), either cherry pick some other policies for happiness bonuses or build Stadiums.
