Here you will find a list of Asian airline carriers. Check below for fares to and from your country. Also flights within Asia are generally cheap. I spent quite a bit of time flying around Asia. Here you can see how much I payed for flights and visa runs.
Be sure to check the countries visa information before purchasing a ticket. All of the below airlines fly within Asia. I have just organized them by their country of origin.
www.singaporeair.com/ One of the nicest airlines that I have ever flown with.
www.china-airlines.com/ I flew with this company around 5 or 6 times back and forth from Hong Kong and Taiwan.