You can begin doing daily quests on Belsavis and start receiving Daily Commendations before you finish your class quest, if you wish. The Belsavis series is given my an NPC outside the shuttle on the orbital station.
When you follow your class quest to Ilum, it will begin a 1-time chain that grants 3 Daily Commendations per quest and you'll be able to do all the dailies along the way. I do not know if you can start Ilum dailies before your class quest takes you there.
The quests I got were probably just lead-ins and I would bet you can do them before the Class quest, if you're able to do Belsavis without completing them. Would be a little odd to have Ilum require completed class quests and Belsavis not.