Pagi, Divjan. I have lived in Indonesia for the past year and a half, what an amazing country! The Hyatt is probably your best bet, the MO closed about a year ago for renovations, I don't think they have reopened yet. Cheaper, but ok, I would do Pan Pacific or Le Meridian (Pan Pacific better location).
I just got back from the Gili Islands about a month ago. Spent a couple of days in Bali, flew to Lombok on Trigana Airlines (not as bad as I envisioned), and then did a boat transfer to Gili T. If you are younger, and like a party island, you would love Gili T.
My favorite place I have seen so far? By far, Lake Toba, in Sumatra. Fly JKT to Medan, and then a four to six hour drive (depends on if your driver is race car trained

. The place was clean, cooler, litter free, no pollution, quiet, and one of the most beautiful lakes I have ever seen. No 5 Star, but neat little places to stay.
I will fly most Indo carriers, however, most of my colleagues stick to Air Asia, Garuda, and Merpati.
Feel free to private me if you have any questions.