MMORPGs have become a lot more accessible over the years, but they can still be very daunting to new players -- particularly if you've never encountered one for yourself.
Final Fantasy XIV actually does a pretty good job of walking you through the early stages of the game, with pop-up help windows and straightforward quests guiding you through your first steps in A Realm Reborn.
What's worth noting is that the game doesn't really open up fully until you reach level 10 in whatever class you chose to start off in; once you have reached level 10, however, you're then able to freely change class whenever you like simply by switching your main hand weapon. This also gives you access to the "Disciples of the Hand" crafting classes, the "Disciples of the Land" gathering classes, the Guildleves mini-quests, the Guildhests multiplayer co-op challenges and a whole lot more besides.
But what should you do prior to reaching level 10? Here's some activities to get you started.
Follow the main quest. This isn't Skyrim; you don't need to try and do as much other stuff as possible before you start on the path to beating the main story. The main quest is one of two ways that Final Fantasy XIV provides you with some well-paced, level-appropriate content -- and it's also where the most interesting, exciting story setpieces occur, too.
Don't forget your class quests! Early in the main quest, you'll be introduced to your starting class' Guild, and every five levels thereafter you'll get a new quest from them. These are particularly worth doing, as the way they're structured tends to be a good means of learning how to use your class' unique abilities and game mechanics in context. Some abilities can also only be unlocked by completing these quests.
Go hunting. Towards the start of the game, you'll be given a Hunting Log for your class, and if you change your mind later you'll get a new Log for each class. Defeating the specific enemies listed in the Hunting Log is a quick route to some massive experience bonuses, so don't neglect it. Enemies you're yet to complete the Hunting Log entry for have a special icon over their heads when you encounter them out in the field -- it looks kind of like a half-eaten cookie for some reason. Assuming you're around the right level, take them down while you know where they are.
Help out other players. Unlike many other MMOs, you can freely attack enemies that other players have already "tagged." You'll even get experience if you help them defeat the enemy, with the exact amount depending on how much damage you contributed in total. Enemies defeated in cooperation with other players count towards quest objectives and Hunting Log entries, too, which helps minimize the need for camping.
Answer the call of FATE. Full Active Time Events are public group challenges that crop up in the world every so often. They show up on your map, and you'll also be notified when you're near one. Jump in and participate and, assuming you contribute a decent amount to whatever the task is, you can find yourself with some excellent rewards, even early in the game.
If you find yourself unsure of what to do next, be sure to make use of the Recommendations system in the menu, which recommends quests, FATEs, hunting log entries and other challenges in your current zone. Don't be afraid to use the Shout channel to ask for help from other players, too -- just type /shout then your message into the chat box and everyone in the area will "hear" you.
Before you know it, you'll be at level 10, and that's where things start to get really interesting. But that's a story for another time.