What should I do/know when I arrive at the airport?

Wifi at the airport is currently only accessible if you have a phone number. If you want someone to lend you a hotspot, that's an option. You will get gouged if you take a ride with one of the unofficial taxis, so walk along the road by the entrance (in the direction traffic is going) and find the taxi line.

Otherwise, find the Metro (line 10) which can, by way of some transfers, get you anywhere you need.

I need a SIM card

Yes, but not every mobile store can get you one. You will need to go to a main office of China Unicom or China Mobile. China Unicom's main branch is located on Kehua Beilu, and offers many different plans. Don't forget your passport.

Where should I stay / live?

Yulin for older, cheaper apartments in a quieter part of town, but not without its share of bars. Many foreigners live in Tongzilin, thus it has the highest concentration of non-Chinese food if that's what you need. These two neighborhoods in the south-central part of Chengdu are popular for foreigners; Jiuyanqiao and Guangfuqiao also deserves a mention.

I need a bank account / debit card

Without a bank account, you are cut out from the WeChat and mobile payment ecosystem. China has adopted these mobile payment systems more aggressively than anywhere else, so while you can survive with cash, it is become less and less convenient to do so.

How to go from A to B?

Taxis are very cheap, are relatively trustworthy. The subway system is good and only getting better, and it is your only hope if you're stuck in downtown during rush hour. If you have a phone, phone number, and bank account, get your WeChat wallet working and download Didi and whichever shared bike app hasn't gone bankrupt yet.