You need to be a blacksmith to make a sword, and you probably have to be a carpenter to make the grip. Seriously. So you'll need to take up mining and botany to get the raw materials. I would just buy one from another player's retainer in the Market Wards at this point. Remember to use Item Search. If you need help finding gear and recipes try ffxiv.yg.com. Another very good idea would be to shout or try the individual world forums to get into a fun LS. Players who have been playing since release often have a bunch of gil and materials and would probably easily be able to supply you with early-game gear.

The tutorial 'quest' is a guildleve. You can grab a new set of 8 regional guildleves every 36 hours. Type /clock or check the Timers tab under Attributes in the Main Menu to see how long until the next guildleve reset. When you first equip a weathered pickaxe, you should learn the ability Lay of the Land, which tells you the direction and distance of the nearest mining node. You can equip this and other abilities by either typing in, for example, /eaction 2 "Lay of the Land" to equip it in slot 2 of your action bar. You can also use the Actions & Traits menu to change abilities. Note the drop-down menu to select a class from which to select abilities. You may also want to set up macros using the /eaction and /equip [slot] [item] commands to make switching classes quick and easy. I have to use five macros to set around thirty-five abilities and traits and ten pieces of equipment when I want to switch to my R50 Conjurer.

Storyline, Grand Company, and Class quests and sidequests unlock whenever you reach the appropriate rank in the required class. You can tell when an NPC has a quest for you because they'll have a '!' above them. You can't skip quests by ranking up too fast, but you must do most quests in order. Main story quests require that you attain a minimum rank in any class, from rank 1 to rank 46. The quest lines differ by city, but merge around R20. Quests for each class unlock at the corresponding guild at R20, R30, and R36, but require that you complete some of the main scenario quests first. Grand company quests first become available when you have any discipline of war or magic at rank 22 or higher I believe, and more will be added in subsequent patches.

What you hear about stats is true, to some extent. INT definitely affects spell and magic attack damage, but not as much as the magic potency of the equipped weapon. I imagine the situation is similar with STR and attack. However, this is planned to be fixed in...1.19 I think? Hopefully we'll see it by the end of this month or shortly thereafter.

That was a lot of text, but I hope it helps!