Grand Company quests become available at I believe Rank 22.

As far as mid-level content there is currently a Rank 25-30 dungeon called The Thousand Maws of Toto-rak that is pretty fun, is some good XP, and drops some nice gear for that level. The higher rank dungeon is 45-50 so there is a bit of a gap, but the coming content (Caravan Escorts, Beastmen strongholds, and the Ifrit battle) should hopefully help bridge it.

Currently there are not many XP groups because XP grinding outside of guildleves is not as rewarding. The patch coming around the end of the month should fix that with the implementation of XP chains, link bonuses, and XP award revamping in general. I'm not sure how it is on your server but usually on Istory there will be a few shouts for mid-level party grinds on days when leves don't reset. You might be able to catch some if you wait around in Ul'dah near the front steps. If you find a good place to grind, you could always just try to start a party. Unfortunately many people don't use the current party search function for various reasons, so you will most likely have to rely on shouts for now.