A city's area of influence measures 21 squares, a large square of five by five squares centered on the city, excluding the corner squares. The city may work no squares beyond those. However, a single square may be worked by only one city; in the city window's map, squares being worked by other cities (or your nation or another), or occupied by enemy units, are marked in red, and you may not place workers there. You work squares by allocating citizens to them.
The square in the center, where your city resides, is automatically worked to produce food, production points, and trade, without the allocation of citizens to work it. A city's center is considered irrigated for this purpose but not mined; further irrigation may not be extended from it. A city does have roads (and railroads, when you have such knowledge) automatically. A city always yields at least one production point, even if the terrain does not allow it.
The Freeciv server has algorithms that determine an optimal utilization of the squares around a city; those algorithms are used by the AI players. Newly created citizens are automatically assigned to squares by those algorithms, which generally favor maximum food production. You may have other criteria for your city, and may wish to reassign workers to other tasks. For instance, you may be aware that the city will have morale problems if it grows too quickly, and may wish to move workers from growing food to extracting ore.
You use the city's map window to move workers off the land, which makes them entertainers (the initial, default specialist type). Clicking on a map square places a citizen at work there, and updates the map to show what yield is expected in food, production, and trade points. You may have to try a number of different squares to decide on the best placement of your citizens.