Welcome to the forum!
Two guides you might like to read:
Early-Game Concepts for Beginners, and
Complete Guide to Happiness.
Looking at that screenshot, it looks like you're doing really well to be honest. You have many cities, presumably most of them annexed, and you've still got positive happiness.
To improve the situation, you could try building/buying courthouses in the annexed cities; connecting up any luxury resources if they aren't already; building/buying other happiness buildings (eg, Colosseums).
It's possible/likely that with such a large empire, as soon as you become happy your cities will start to grow again, pushing you back into unhappiness. To fix this you could choose to prevent city growth - it is an option within the city screen over by citizen management.
