Kaitlyn,
In my experience having Bangkok as the end destination results in the cheapest flights. However, I usually take advantage of the one stopover permitted for such long flights, which allowed me to stop in Japan on one trip and China on another trip for no extra cost other than the in country expenses.
My recommendation for your itinerary, once you get to Bangkok and have spent a few days touring this crazy city, would be to take a flight to the southern islands for a beach vacation to recouperate (Bangkok to Krabi..the gateway to the islands on the Andaman Sea is $60 on Air Asia). There you can go to Ao Nang, Railey, Phi Phi, and Phuket. Then take the Air Asia flight from Phuket to Chiang Mai for $46. Look at other blogs for what to do in Chaing Mai (e.g., jungle treks, etc.). There are many opportunities to volunteer at "orphanages" in northern Thailand, as you mentioned you wished to do. Let me know if you need contact information.
Then look at the blogs to see if you are up for the overland route via minibus and boat, to Luang Prabang, the best place to see true Laos (most describe it as a trip to hell). Otherwise take the Lao Aviation flight for $176. I would take the shorter trips in Laos in minibuses to places like Vang Vieng...if you are into that scene...again look at travelblog for other Laos destinations. I would skip Vientiane in favor of more time in Luang Prabang. The air fares from Chiang Mai to Luang Prabang to Hanoi are pretty steep as they are not on discount airlines. To get the discounts you will have to fly back to their hubs and then out again to these destinations (into Vientiane for best prices).
Then look at the blogs to see if you are up for the overland route to Hanoi (also described as a trip to hell). Otherwise take the one hour flight from Luang Prabang to Hanoi on Vietnam Airlines for $200. In Hanoi, make sure you book the three day tour to Halong Bay, and don't pay for the cheapest tour. I'm not sure I would go to Sapa, but you can research the travelblogs related to Vietnam destinations to see what appeals to you. Some don't like Dalat, but it is a cool break, and I'm partial to that city having lived there for eight years. I would take either the sleeper buses or the train down the length of Vietnam, stopping at those places you have identified, and ending in Ho Chi Minh City (HCMC)/Saigon.
There are buses or bus/boat combinations from HCMC/Saigon to Phnom Phen, and from there to Siem Reap/Angkor Wat. Cambodia is also a good place to volunteer. Again, let me know if you need contact information.
I would then take the Air Asia flight from Siem Reap to Kuala Lumpur (KL) for $96, even though you didn't list Malaysia as a destination. However, you did want to go to the Philippines and Indonesia, and KL provides the cheapest flights to both countries (KL-Bali R/T for $143, and KL-Clark/Manila for $237 on Air Asia), assuming Bali as your Indonesia destination. Also, Malaysia is worth visiting...I like the Cameron Highland (another cool climate spot where I lived for 2 years), and Penang...take buses within Malaysia...all shorter distances. As for open jaw tickets, flying a discount airline from Bali to Bangkok ($99 on Air Asia) is probably cheaper than flying from Bali back to the States. However, you might check the cost of flights to Bali with a stopover enroute in Bangkok. Sorry about the length of this reply! Bob
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