You may find that your capital has enough wonders and stuff in it that you don't need to build the Cathedral-class buildings in it - just the "other two" cities. I'll usually make my capital one culture city, and it generally has enough culture that it'll get legendary status at the same time as the other two without any special attention. Second will be a science city if my capital is a gp farm/money/religious city, or a gp farm/money/religious city if my capital is a science city. Third will be just a culture city, where I'll build globe theatre and hermitage and all the cultural improvements I can find.
If I'll only need Cathedral-class buildings in the second and third cities, I only need three additional "minor" cities for that. You'll probably want at least four "minor" cities, however, so you have enough universities/banks/theatres to build Oxford/Wall Street/Globe.
Eventually, when you get the techs to get the buildings you want in your three cities, you switch the sliders from science to culture, and make any specialists in the cities that need to build culture artists. This can be done early, but I like to do it after getting biology (for better farms), and electricity/radio/mass media (for the entertainment wonders and Eiffel tower)