The bonus series starts when you finish the main story of Nar Shaddaa, along with your class quests located there *and* you are of appropriate level (not sure what level is required yet).
According to OP he is 29 and ran out of quests on Alderaan. I would suggest the following:
- gather the group for daily instances: mandalorian raiders, cademimu, which ever is of green or yellow difficulty gonna be fine for your experience - plus you will get some gear drops
- start queueing for warzones if you find pvp fun, along with completing the daily pvp quest
- start completing the space missions (at some points investment into ship gear will be necessary)
In my experience Alderaan isn't designed well from questing side. I found the story incredibly good but the quests alone, their objectives and amount of travel needed was just a bit too much of a hassle. I cant say I had troubles with level there.
Good luck OP and please consider completing flashpoints, warzones, space missions - whichever suits you - to get some additional exp.