Now that you have the lay of the land, you have to keep the other player from getting their grubby little hands on it! You will do this by taking turns adding tokens to the board until only one player's territory remains (or the other player resigns at the realization of impending, crushing defeat). Each turn consists of 4 phases: claim, subsume, conflict, and movement.
Players begin on opposite sides of the board. To determine who goes first, one player puts a token of each color in their hands and the other player selects one without looking. White begins play.
At the start of their turn, each player gets a 1u token to place on the board. They may put it either on an empty square on the row closest to them (their "home row"), or on any existing territory they have. Placing the token on an empty square is said to "claim" it for the player. Placing a token on another territory of theirs increases their claim to the area, stacking up tokens and eventually leading to "control".
A territory is under control when the number of additional tokens is equal to the length of its uncontrolled border. For example, 5 1u tokens control a 1u area (1 for the initial claim and one for each edge), 12 for a 2u (4 claim, 8 edge), 32 for a 4u (16 claim, 16 edge), etc. In order to control larger territories, you must join smaller ones.
When 4 controlled territory squares are arranged next to each other such that they can form a larger territory square, the player can choose to subsume the smaller territories into a larger one. Doing so frees up tokens, since a larger territory can be controlled by fewer tokens. For example, 4 1u territories take a total of 20 tokens to control, but the 2u territory that subsumes them requires only 12, freeing up 8 tokens for other uses. Note that having less than 12 tokens does not mean that you lose the larger territory, only that you lose control. You do lose the territory, however, if you don't have the minimum number of 1u tokens to claim the entire area.
Conflict resolution is discussed in detail here.
Finally, at the end of a player's turn, any 1u token on the board that has not already participated in conflict is then allowed to be moved to an adjacent empty square. Keep in mind, however, that stack ordering still matters, so if you move a token from one territory to another it completely prevents any tokens on the second territory from moving that turn. Movement is only horizontal and vertical, so movement to a diagonal square would require two turns.