Almaty is a city with a mountain wall at the end of every street. The Soviet planners laid it out on a tidy grid in the 1920s and 30s, when it became the capital of Soviet Kazakhstan, and it stayed the capital of independent Kazakhstan until 1997, when the government moved to Astana (3 Days in Astana). Before that it was Verny, a Russian fort town founded in 1854 that two earthquakes, in 1887 and 1911, nearly wiped out. The name is thought to come from the Kazakh word for apple, alma, and the wild apples of these hills are believed to be the ancestors of the apples in your fridge.
It's not a pretty city in the way Prague is. It's a working, slightly rumpled one with wide avenues, Soviet concrete with real ambition, a wooden cathedral that looks like a cake, and good cafes. What makes it worth five days is the mountains and what's around them. In an hour you can be at 2,200 m at a ski resort, and in half a day at a canyon that looks like Utah. Five days gives you the city properly, a day at Medeu and Shymbulak, a day at Big Almaty Lake and one long day out to Charyn Canyon.
This plan is written for November, the shoulder season when the lifts are being serviced, the rink is shut and the first snow falls on the peaks, and it says plainly what is open and what isn't. Day one is the wooden cathedral, the park and the Green Bazaar, day two the Soviet city and Kok-Tobe, day three Medeu and Shymbulak, day four Big Almaty Lake and day five Charyn Canyon. Start on a Tuesday, because the Green Bazaar's main pavilions are closed on Mondays (the Central State Museum is listed as closed on a Monday or a Tuesday, depending on the source, and neither is day two). Kazakhstan is a big country, and for the steppe, the nomad food and the rest see our Kazakhstan page. In summer the same plan runs fine, with a lot more open.