Well, you definitely should get good internet in Singapore.
If you're doing Bali, the whole Kuta (beach) scene is pretty overrated and not even close to as good as it's hyped. Also: most of the beaches in Denpasar/Kuta/Airport areas have major garbage issues. If you want to do some really great cultural sightseeing, head to Ubud. Lots of great Hindu temples, the old Palace is there, and the best suckling pig you'll ever eat. In fact, IF you do go to Bali, I know the number of a GREAT driver you can hire for $30/day. He knows the island like the back of his hand, but there's a chance he's already hired. PM me if you want his number so you can SMS him.
Other than Bali: Jakarta is in the middle of the worst flooding it's seen in 5 years, but if you make it there after the water recedes, it's probably the one city in the entire world where if you have enough money, you can do whatever you want. There is no limit. It can get pretty sketch from what I hear, though. Not for anyone who doesn't mind getting their elbows dirty in order to have a good time.
I can't imagine there's too many other places you'd want to go / that are well known internationally. If you're doing a "culture" trip, Yogyakarta is pretty great. Borobudur is there, it's a "can't miss" temple if that's your kind of thing.