You can still grind in ARR. I guess that just depends on how strictly you want to define grinding though. If you're thinking grinding as in the FFXI version where parties would go to a camp and pull mobs for 5 hours straight for minimal xp gain per mob, then no, there is no more grinding like that in FFXIV.
However there are plenty more ways to gain levels in ARR. It's not just quests and dungeons, though they are both a good source of xp. You can actually go out, make a party, and grind mobs. Only the mobs you'll be targeting are off of your Hunting Log. These make great xp and generally require each member to kill between 5 and 10 of each type. You can do levequests. Let your levequest allowance fill up for a day or so and have fun burning through 20-30 of those suckers. That'll give you the ol' grinding feeling right there. There are FATEs which give great xp as well. Guildhests, which give amaaaaaazing xp whenever there's someone going through them for the first time in the party. These also have the added bonus of teaching very important party mechanics necessary to complete later dungeons. There are of course dungeons. You can spam these for levels and gear. And they're fun!Â
So yeah, not everything in ARR is quest-based leveling. You've got options. And I think these alternative ways of leveling would be a lot easier to RP than sitting around grinding mobs and trying to make that interesting. But that's just my opinion.
However there are plenty more ways to gain levels in ARR. It's not just quests and dungeons, though they are both a good source of xp. You can actually go out, make a party, and grind mobs. Only the mobs you'll be targeting are off of your Hunting Log. These make great xp and generally require each member to kill between 5 and 10 of each type. You can do levequests. Let your levequest allowance fill up for a day or so and have fun burning through 20-30 of those suckers. That'll give you the ol' grinding feeling right there. There are FATEs which give great xp as well. Guildhests, which give amaaaaaazing xp whenever there's someone going through them for the first time in the party. These also have the added bonus of teaching very important party mechanics necessary to complete later dungeons. There are of course dungeons. You can spam these for levels and gear. And they're fun!Â
So yeah, not everything in ARR is quest-based leveling. You've got options. And I think these alternative ways of leveling would be a lot easier to RP than sitting around grinding mobs and trying to make that interesting. But that's just my opinion.