Once you have established your city you will want to expand your population so you can make more money. To achieve this will require that you raise your city's rank. This is done by acquiring a certain number of points which are attained by collecting trophies which are earned by completing achievements. You can always view the trophies and their associated challenges by tapping the 'Trophies' icon in the upper left corner of the screen.
When you have the specified amount of points you then need to complete a challenge that involves a face-off between your buildings and the AI's. The winner of this face-off is largely determined by land value so ensure you know the value of your land before challenging the AI.