In terms of your starting area, roughly all of them are the same, and offer up similar beginner quests to acclimate you to the basic mechanics of A Realm Reborn. Where you start depends entirely on your class (not your race), so your choice is essentially a desert (Ul'dah), a sea-faring town (Limsa Lominsa), and a forest (Gridania). Very early into the story you'll unlock airship travel and jump between all of three of them anyway, so just pick your favorite starter theme. If you're curious, I think Limsa Lominsa is probably the easiest town to learn the layout for, as it mostly consists of straight hallways. Ul'dah is a giant circle with multiple snaking hallways and Gridania is separated into two major "blocks."
Don't get too intimidated by the massive cities at first -- there are multiple ways to level up. The easiest way is to follow the main quest (denoted by a starburst around the typical quest icon on your map and compass). The core story will inherently bring you to new areas and hubs, at which point you should grab every single quest in the area and do them immediately before moving on. This prevents you from being underleveled, which can be a problem if you get into an area that's too high for you and all of the quests are locked (denoted with a key icon on the bubble).
You can also participate in FATEs (Full Active Time Event), and I recommend you do so at every turn. Every so often on your map you may see a purple star icon or a blue bubble, which denotes that a FATE is happening. Hop in there and complete the objective, which is usually just "kill these enemies/boss" the vast majority of the time. Depending on your contribution, you'll earn a certain amount of experience, and the bonuses are massive enough to go out of your way to do them. If you are too low level, your experience boost won't be weighted as high, and if you're too high level you can sync down to the appropriate setting by using the button in your quest log. Pay attention to "FATE Groups" in chat or the "looking for party" function, as they tend to just go from FATE to FATE and grind experience.
Another way to circumvent the underleveling problem is by way of Levequests. These are repeatable quests that allow you to do the same objectives over and over until you run out of "allowances," which are viewable on your quest log/journal. Look for these in nearly every town, denoted by a playing card symbol on the map. All you have to do is take the quests, and after a certain amount of time you'll earn now allowances (check the "timer" option in the main menu). If all else fails, you can take a break from the game for a few days to stack up some allowances as well as bonus rested XP (more on that in a second).
Yet another way to assist in the leveling process is the duty finder, unlocked after you reach level 10. You can do a lot of different things here, including dungeons (full instances with both regular enemies and bosses), Guildhests (small easy quests usually only involving a few encounters), and Trials (challenging encounters with just one boss). When in doubt, progress with the story as far as you can to earn the most appropriate dungeon for your level, and clear it a few times to get past any ruts you may be having. You can also opt to do a "roulette" once a day for bonus experience.
Finally, you can rely on rested XP to help increase your experience gain when killing foes (it won't boost quest XP though). All three major towns have an Innkeeper that you can visit to retire to your room and earn an extra rested experience pool. If you're going to log out for an extended period of time, head to the inn first. The easiest Inn path is to teleport to Limsa Lominsa, teleport to the Aftcastle, and run backwards into the Drowning Wench alehouse. As you can clearly see, you always have options when it comes to leveling your character up to continue on with the story.
When adventuring, make sure you keep a look out for every single Aetheryte crystal in a given zone. Every zone has at least one, if not two crystals, which you can use to later transport yourself around the world for a nominal fee. Also do yourself a favor and grab every Chocobo porter in each area, denoted by a person with a Chocobo hat and an exclamation mark above their head. Now that you understand the basics, here are all the major events you'll experience up to level 30, which I think of as monumental scenarios.
Around level 10-15 depending on where you are in the story, you'll have the opportunity to change classes. You can power through any class you want to progress with the main story, but note that any quests you do will not be available again on that character -- that's where the extra weighted XP for your other classes comes in handy. Select classes are highlighted in your actions and traits menu, which denote the ones that can gain access to cross-class powers -- those are the first ones you may want to level. For instance, the Lancer has the power to use an ability from the Marauder class to leech health while attacking, and heal themselves instantly with a power from the Pugilist class.
At level 20, you can join a Grand Company (The Immortal Flames, The Order of the Twin Adder or The Maelstrom) from any one of the three cities. There's a very small difference between them, mostly having to do with PVP. The fact remains that you'll still be able to get every main and sidequest from any Company, so don't think of it as a faction that will cut all ties with other options.
The main benefit from joining a Grand Company right off the bat is the power to embark upon your mount quest. Simply talk to your Guildmaster at your appropriate Company and head to any of the three questgivers they denote in their conversation with you (Camp Drybone in Eastern Thanalan is the easiest in my opinion). Just grab some Levequests and grind them until you have 2,000 credits with your Company (it should only take you 30 minutes if you have the allowances), which you can exchange for an item that lets you gain access to your first Chocobo. This also lets you ride any of the mounts you may have picked up through other means like buying the Collector's Edition.
Once you're level 30, there's a huge event on the horizon -- you can change into your core job, which is basically an upgraded version of your class. Talk to your class trainer at 30, and note the other class that you need to level up. Using the Lancer as an example, to get the Dragoon job, you need to be a level 30 Lancer and a level 15 Marauder. It will be much easier to level your second class because of the bonus XP, and it should only take you roughly an hour or so to get level 15. Complete your original class quest from your trainer to gain access to your job, as well as at least one new ability.
From level 30 on look for Leves, make an effort to do FATEs, do the closest dungeon to your level, and complete every quest from the duty finder at least once to earn bonus XP. If you're ever stuck at any time, just remember the fundamentals of gaining levels. If you're out of Leve allowances, you're locked out of a story quest, and you can't seem to find any FATEs, queuing up for the duty finder is always an option. You'll have a few slow levels in your 40s, but you'll be hitting endgame before you know it, and by then you'll have memorized the world map.