How to begin a new game of Final Fantasy XIV after creating a new character.
Beginning the Game
After you have chosen a datacenter and server, and after you have chosen and confirmed the name for your character, you can begin a new game. You may need to wait in the login queue first. When you log in for the first time with a new character, a cutscene will play. After the cutscene, there will be some dialogue.
Choose a Control Scheme
You are asked if you will use mouse/keyboard controls, or controller controls. If you choose controller, you can press Start to access the main menu. If you choose mouse/keyboard, the main menu buttons are in the corner of the screen. You can switch between mouse/keyboard and controller at any time via Character Configuration in the System menu. For a more detailed explanation of the controls, see the official site's Basic Controls page, which includes a list of keys/buttons and what they do.
Complete the First Quest
Near where your character starts, there will be an NPC (non-player character) nearby with an exclamation mark ! above their head. After you have familiarized yourself with the basic controls (which you can review by choosing Active Help in the System menu and choosing Basic Controls), talk to that NPC, read the information that appears, and use the Accept button to accept the quest. A cutscene will play.
After the cutscene, your next destination will be marked on the minimap. You can click the minimap, or choose Map in the Travel menu, to open a larger map if needed. Another way to find the quest objective is to look at the list of your current quest(s) at the side of the screen. You can click a quest objective to open the map and show the location of the quest objective.
Follow the quest objectives until you reach an NPC with a green check mark above their head. Talk to them. Another cutscene will play. The Adventurers' Guild leader will give you some information about the city that you started in. They will also mention the looming threat of invasion by the Garlean Empire, and will refer to the Calamity, a disastrous event that occurred five years ago. (You can watch the End of an Era cinematic to see this event. For more background information, you may wish to watch the Noclip FFXIV documentary.)
After the cutscene, click the Complete button to complete the quest and receive your rewards of gil (the in-game currency) and experience points. This completes the tutorial phase of the game, and your next step is to talk to the NPC again, who now has another exclamation mark ! above their head. This time, the exclamation mark is inside of a meteor symbol, which means that it is a Main Scenario Quest.
If you ever can't find the next or current main scenario quest, you can click the quest name under "Next Main Scenario Quest" or "Current Main Scenario Quest" in the corner of the screen to display it on your map. If you don't see this in the corner of your screen, go to System in the main menu, choose Character Configuration, go to UI Settings, and select the HUD tab. Scroll down to "Display Main Scenario Guide" and make sure that it is checked, then click Apply to save your changes.
Talk to the leader of the Adventurers' Guild to start the next quest, Close to Home.