There are plenty of 'EXP' boosts in the game as well as leveling opportunities very early on. The problem is, the game itself doesn't point them out.
For any level there are the basic story and sidequests in your starting city, guildleves, guildhests, fates and hunt log (which is attached to your chosen class). Once you hit lvl 15-16 you can do the Smiths training which will net you a ring that will give you an exp boost to level 30, unlock sastasha for leveling and also unlock Deep Dungeon which is a preferred method of leveling which also gives you the chance to earn rewards and items (you have to have access to the physical area deep dungeon is in but still).
There's food items you can buy from any vendor that will also boost your earnings, certain clothing sets in the cash shop that offer an exp boost to certain levels (that can be paired with with other booster jewelry), special fates that boost your exp later on, not to mention the booster earrings you get simply for buying the expansions.
Can you level by simply mowing down enemies on the world map? Yes but that is actually the slowest method. The game is paced so that you level as complete story. Which is important to understanding the gamenas a whole.
Those games that let you level quickly usually have a huge gap between their casual content and more advanced content that requires a lot of real cash in the form of statboosting items or a lot of grinding for said items. Gear is pretty straightfoward in FF14. You can participate in any advanced duty you have the ilvl for. Its no guarantee that you'll clear but you can.
Seriously, leveling is only slightly less fast on FF14. I know someone who leveled to 80 in 3 days. Four hours to level to max in a game is ridiculous for a new player. The system could be better here but it ain't that bad either.