Note that because Hong Kong is such an international city with a thriving spirit of market-capitalism, flights from America, Europe, etc to Hong Kong are almost always cheaper than flights directly into the PRC mainland. However if you must overnight in Hong Kong the advantage is lost because you will spend a small fortune for a hotel room. Add to that the high price of the Hong Kong-Chengdu leg, and you erase any price advantage obtained by going this way.
Therefore the absolute cheapest way to go from, say, San Francisco, is to fly on a consolidator ticket to Hong Kong, then go from the Hong Kong airport immediately to... (1) the Canton ferry terminal, catch an overnight ferry to Canton (also called Guangzhou). When you land in Canton get a taxi to the airport, and get on a plane for Chengdu. OR (2) the Hong Kong KCR train station. Get on a train north, there are departures about once an hour. When you disembark at the end of the line, walk across the border to Shenzhen, get on a taxi to the airport, and catch a plane to Chengdu.
Note that all flights between cities within the mainland should be booked in advance, because they are often full. As far as we know, there are no discount tickets available for air travel within China.
It is even cheaper if you travel across China by rail, and not a bad way to see the countryside. Consult a guidebook such as Lonely Planet, or a travel agent.
The middle path, striking a nice balance between comfort and cost, is to transit in Beijing or Shanghai.
The most comfortable and convenient way is to fly to Hong Kong, overnight there, then fly on to Chengdu on Dragonair . This carrier is head and shoulders above any mainland airlines in service and safety. It does cost more, however, and flies to Chengdu only on three days a week (including Sundays).
A couple of important notes about Hong Kong: (1) Handover not withstanding, citizens of most western countries still do not need to obtain a visa in advance to visit; and (2) This is the quickest place to get a visa for the mainland. If you don't mind paying an extra fee, you can get your Chinese visa in as little as three hours here.
Other air routes to Chengdu include Lhasa-Kathmandu-Chengdu or Bangkok-Kunming-Chengdu or Singapore-Chengdu. These routes are not served every day. Transiting in Lhasa requires a special permit which may be difficult to get.