Although it's not the capital, Almaty is a great city to explore. There's multiple attractions to check out and majority are within walking distance of one another due to the compact layout of the city. The highlights being Central State Museum, Glory Memorial, Ascension Cathedral, and First President's Park. I found Almaty city itself an enjoyable city both in summer and winter!
The Big Almaty Lake is a popular spot for both tourists and locals due to it's close proximity to Almaty Centre. Within 30 minutes you can be in the mountains having a picnic next to Almaty Lake. The landscape en route is beautiful with breathtaking mountains and greenery for miles (season depedent). This isn't a lake you can swim in as it is a protected area, but very rare you'll find beauty like this so close to a populated city.
Charyn Canyon is definitely worth the 3 hour drive East from Almaty. The canyon is located on the Sharyn River and is roughly 90km in length. Charyn National Park does appear on googlemaps so is relatively easy to reach. Some of the roads en route are not the best and you may find you lose GPS signal on multiple occasions, so best to go with a local if possible. Once you reach the National Park, everything is signposted well and there's a newly built car park.
I've seen many canyons over the world and can verify that this one is truly stunning and really doesn't get the praise and recognition it deserves.
A 5 hour drive East of Almaty will bring you to the beautiful Lake Kaindy. Yes it's a long drive, but a day trip to the lake can be combined with Charyn Canyon. The lake is 400 meters in length with a depth of almost 30 meters and is very unique. A landslide back in 1911 which formed a natural dam is essentially how the lake was formed. Over the years, this dam has been filled with mountain water, which is why the lake is often referred to as "The Sunken Forrest" due to many trees poking out of the lake. Easily reached and requires only a short 20 minute walk once you reach the park and enter the barrier.
Kazakhstan also has a ski season during the early months of the year. Okay, so it's definitely not a comparable to European resorts of France and Switzerland, but for a low cost ski trip within Central Asia, Shymbulak Ski Resort is a winner. Another huge bonus is that the resort is only a 40 minute drive from Almaty Airport, so the transfer time is minimal. There's various lodges in the area, but Shymbulak Resort Hotel is by far the best due to its location directly on the slopes. You're looking at $40 USD for a day pass and gear rental, so a fraction in cost when compared to others areas of the globe.
Located within close proximity to Shymbulak Ski Resort is Medeu Skating Rink, which holds the prestigious record of "highest skating rink in the world". More of a popular spot in the winter months due the the high speed skating facilities on offer. The complex opened in 1951, but has been renovated on three separate occasions since, with the most recent being in 2009. The venue, along with Shymbulak Ski Resort was used for the 2011 Asian Winter Games, which is still the largest event ever held in the country.
Almaty also has an impressive Opera House with numerous shows on throughout the week. Many ballet shows are also available via the booking site. As ticket prices go, this is a very cost-effective country to watch a show. Worth noting that the performances are in Russian language, so a ballet show may best suit. You can find the booking website by clicking here!
There are multiple hotel options within the city of Almaty, so I thought I would recommend an option for ski purposes. Shymbulak Resort Ski Hotel is a clear winner due to its location directly on the slopes. There's multi room options available for around $70 a night upwards. The facilities are basic, but all rooms have en-suite bathroom and a direct view of the ski slopes.
Olivier Restaurant & Bar in Almaty would definitely be my restaurant recommendation for your trip. A great dining option in the centre of the city offering both local and international dishes. Mid-range in the price bracket and also has an extensive range of alcohol options compared to elsewhere in the city. Outdoor seating is also available during the summer months.