I have a played a few and this has on of the least worst grinds for leveling the first couple classes then the other MMO's I played. It also sounds like you started at launch which is usually the worst time with the least way's. They have added lots of low level side quest that can help once you are 10+. MMO's are based on endgame. Stopping at level 20, when you just joined a grand company and you just got your mount, does not really make since I mean you either quite before finishing these quest or just weren't paying attention to the story. You haven't even unlocked a job yet, this is when most things become way more interesting. Some jobs don't really get work till near lvl 50. You get 1 piece of a combo at lvl 10 another at 20 then you won't get the last piece on some jobs until lvl 45+. Or they get the job trait/skill that really defines the job.
Once you get level 50 you will be wishing you had both more time and less time at the same moment on the GCD. The reason it feels slow now is because
1) your a low lvl so you don't really have any off gcd abilities and their isn't a lot to be doing while fighting, as the dungeons get harder their will be more stuff going on for you to worry about. their are a few major "jumps" haku manor(29), stone vigil(41), Aurum vale. These dungeons take drastic jumps that require you to think of everything you have done so far in a previous dungeons into one.
2) your playing a job that doesn't have off gcd abilities. Fates offer good xp rewards and grand company seals. If you want a lvl 20 job to be more fun lvl all classes to lvl 15 this will give you access to most of the cross class skills needed. Several cross classes will work off gcd and give you more to do
Leveling: levequest and dungeons are the fastest way depending on your lvl and gear(more so for levequest). Fates give good xp as long as you can get a gold/silver rank and you aren't the only out there. They also give you grand company seals once you have a grand company. Grand company seals buy you above average armor, minions, crafting items, items to enhance xp earned, and items you can sale for lots of gil. The collectors edition(you can buy a digital code) gives you a helmet that increase xp earned from lvl 1-10 which is great.