Chengdu doesn't rush you the way Beijing or Shanghai do, and it has had a long time to practice. It's the only major Chinese settlement that has kept the same name through the imperial, republican and communist eras, more than two thousand years of it, and for much of that time it was the capital of somewhere: the Kingdom of Shu, Liu Bei's Shu Han in the Three Kingdoms period, a string of regional courts. The usual explanation for the city's famously easy pace is a canal. Around 256 BC the Qin state dug the Dujiangyan irrigation system, the floods on the Chengdu Plain stopped, the harvests became reliable, and people were left with a lot of free time.
Most of that free time still goes into tea houses, hotpot and mahjong. UNESCO named Chengdu a City of Gastronomy in 2011, and the Giant Panda Base is the booking to make first. Five days works because the city itself wants slow lunches and a long afternoon with a cup of tea, and because the two best day trips, Leshan and Dujiangyan, each take a full day. One honest warning: the sky is often grey. Chengdu has one of the lowest annual sunshine totals in China, with only about 15 percent of possible sunshine in December, so don't plan around views.
The plan keeps each day in one part of the map. Day one is the old center with Wenshu Monastery, People's Park and Kuanzhai Alley, day two the pandas, the museum and a proper hotpot night, day three the Leshan Giant Buddha, day four Dujiangyan and Mount Qingcheng, and day five the Three Kingdoms, a poet's cottage and Sichuan opera. Mount Emei is under More time, with the reasons. Start on a Monday: the Chengdu Museum is closed that day, and Leshan and the panda base are busiest at weekends, so this puts both mid-week. Entry rules, payments, internet and maps are the part that needs doing before you fly, so read Before you go first, and the China country page has the background.