1) Go to Medeo and Chimbulak! Medeo is an outdoor speed skating and bandy rink. It is located high in the mountains. From that place you can take Funicular and go to Chimbulak. In winter people usually ski there, but it is very nice at Summer as well. There is a very nice mountain view and some restaurants.
2) The other place is Koktobe. It is also in mountains. You need to take funicular from the city. There are some restaurants and shops as well, but the most important - you can see the whole city from there.
3) If you want traditional Asian cuisine visit Alasha or Samarkand restaurants. They have very beautiful Uzbek style interior and good Asian food.
4) Just walk at the center of the city to feel the atmosphere. It is beautiful and very safe. Younger generation speak English well.
5) If you need shopping, cinema and some fancy restaurants go to Esentai Mall. There is Saks fifth avenue store and all famous brands.
If you have more than couple days there are very beautiful places around Almaty like Charyn canyon, Turgen Waterfalls, Issyk (gold warrior) and especially Tamgali where you can find Petroglyphs. (Google all these places). It is better to go there with your local friends or to take special tour because there is no transportation for tourists. almost 5 years ago