From what I understand, companies are not like kingdom missions in 11. You're able to accept quests from each of the cities grand company with the exception of instance quests. Say, for example, you accept the Ul'Dah grand company quest for Toto-Rak. The quest will warn you that you cannot accept the same quest from Grid or LL. I'm not sure of the minimum level to start banging out quests but before you accept the quests, it will show you a recommended rank.
To start diving in to these quests, head to the area where you would zone to the market wards and choose (at least in Ul'dah) the second option. I haven't signed up with LL yet but Ul'dahs grand company is in the Hall of Flames.
**Please feel free to correct me for misinformation**
FFXIV - Goblin - Seraphan Amatsuka