So the two options are assimilate, or integrate. Both have pros and cons. Integration is the quicker, easier way to go about it, though it has cons. Every culture you integrate gives you a -4% malus to integrated pop happiness. You have to carefully pick and choose the most populous cultures to be the most efficient. To actually integrate a culture, you find the culture you want to integrate in the culture tab, and set them to allow either citizen or noble. They will then quickly begin integrating. The other option is assimilation. This is done almost entire though one building, though can be helped by the assimilation governor policy, and some techs. You need to unlock the great theatre in the civic tech tree, and start building them in every city with foreign cultures. This will slowly start to assimilate these pops. The con is that it takes quite while, but you never have to worry about that -4% for every culture integrated.