Given your time frame maybe you should try Europe instead? It's a much quicker trip travel-wise. Plenty of history, bars etc. And depending on where you choose to go you can get it done in your budget.
But if you are set on Asia I agree that Thailand is a good place. Your budget can take you a ways there, although $2000-3000 total including airfare will mean you need to watch things a little. But you can get perfectly fine hotels for $50-60 per night in BKK or perhaps less depending on your standards. Getting around is not too expensive but when you start adding on trips in-country (to Chaing Mai or wherever) the costs can start adding up a bit.
I don't really know much about costs in Korea but maybe if you just went there you could save some time at least by just taking those direct flights.
There's really nothing magic about booking a connection or multiple flights. I usually check fares using the ITA Matrix tool thing (
http://matrix.itasoftware.com/). But you can also use Kayak.com or Expedia, Travelocity, etc.