Budget Planning

Asia as a region is very large and diverse. From a cultural perspective, India and Nepal are quite different from Japan, China, or Taiwan. From a budgeting perspective, these regions are also quite different. Southeast Asia, India, and Nepal have long been established as "backpacker" destinations and are incredibly affordable and relatively easy to get around. China has been increasing in cost in recent years, and shows no sign of slowing down. Japan and South Korea have always been quite expensive, but the modern infrastructure make these places quite easy to get around.

Things to Do

Scuba Diving: Southeast Asia offers some of the best scuba diving in the world. It is also incredibly affordable to get certified in places like Koh Tao, Thailand. If you're going to be traveling for an extended period of time and haven't done so, consider getting certified while you are in the area. While Koh Tao is a cheap place to get certified, it's not the best place to appreciate diving. It's been beaten and worn down, so if you're looking for a more pristine experience consider going to Koh Phi Phi. Alternatively, perhaps the most famous, but not best place is Phuket.
Trekking: There are many options for incredible trekking throughout Asia. The most obvious and impressive place to do some lengthy hiking is Nepal. There are treks ranging in length from weeks, like the Annapurna Circuit, to several days, like those based out of Pokhara or Kathmandu. If you're in other countries, like China, consider doing Tiger Leaping Gorge, which is an incredible village to village trek. In the area north of Hanoi, known as Sapa, Vietnam has beautiful areas to hike with rice terraces and Hmong villages. In Thailand many people use Chiang Mai as a base for trekking. Essentially, whatever country you visit in Asia, you're likely to find a wonderful place to spend some time hiking.
Festivals: Like in many parts of the world, festivals are a fun way to experience the culture and religion of an area. Some places, like Varanasi, India, have nightly events that draw worshipers and travelers alike. Most places celebrate a once a year festival, like the Naadam Festival in Ulaanbaatar. If you happen to be in a country for a significant festival, definitely get out and about and enjoy some of the activities, but remember the popularity of the event can dramatically drive up the prices of hotel rooms or even food.
Transsiberian Railway: Many people choose to do the Transsiberian Railway as a way to see parts of Asia. It's a wonderful journey starting in Moscow, Russia and ending in Beijing, China. If you choose to do this route, the most affordable option is to buy train tickets as you go. Split the journey up into different intervals. Change trains at the border instead of buying international tickets and you will save a significant amount of money. Also remember that your trip is more than just a train ride. Take long breaks so that you can really see the places you're traveling through.

Budget Travel Destinations

Perhaps the most affordable country you'll ever visit is India. Even large cities like Delhi and Mumbai are quite affordable. A close second is Nepal. Despite it's low cost, you can find some great quality places to stay and visit in India and Nepal. On the other end of the spectrum, Japan, specifically Tokyo, is quite an expensive destination. If you're on a tight budget this is a place you might want to consider couchsurfing. You can find hostels, but they are expensive, and the sleeping conditions are variable. China has been growing in cost in recent years and continues to do so. The quality of accommodation is variable, and often you get what you pay for. The northern areas and larger cities like Shanghai are dramatically more expensive than the south and rural areas near Yangshuo. Since prices continue to rise, don't let this deter you from visiting China now. It will likely only get more expensive.

Transportation Budget

Transportation throughout Asia is quite variable. In China you can experience the bullet train, or you can take the bus. Indian buses and trains are part of the travel experience. For many of the train routes, remember to book in advance because they fill up. Transportation in Japan is expensive and flying may be the best option. Throughout Southeast Asia there are frequent tourist buses, or you can find the local buses that are a fraction of the cost and may or may not be less convenient.