Map from Google - Kazakhstan

Kazakhstan, a vast country of Central Asia, has a markedly continental climate, with very cold winters almost everywhere, while summers are warm in the north and definitely hot in the south. There are no obstacles which could protect the country from cold air masses of polar or Siberian origin, while in summer the hot winds from the deserts of Iran can blow. Central Asia gets very cold in winter and very hot in the summer, because of the distance from the oceans, so that both the highest and coldest records are noticeable: in the north the temperature can reach -60 °F in winter, as well as 105 °F in summer, while in the south it can go from -31 °F to 113 °F. The clashes between different air masses can give rise to strong winds and dust storms, especially in spring and in the south. In the cold months, northern winds can cause blizzards, but without great snow accumulations due to the scarce amount of precipitation.
The distance from the sea is also the cause of the climate aridity, in fact, the country is almost entirely covered by steppes and deserts; the northern area (roughly above the 50th parallel) is the rainiest part, where precipitation exceeds 12 inches per year, mainly because of afternoon thunderstorms which may occur in summer, while in the south-central, it comes down to around 6/8 inches per year, and even to around 4 inches in the area of the Aral Sea. Snow, in the long winter months is quite common, but it's often light and not abundant. There are approximately a hundred days with snow each year on the plains in the far north (see Petropavl), about 60 days in the central region and about 20 in the southernmost region.
The mildest area in winter is the south-west, along the coast of the Caspian Sea, where the average in January goes from 21 °F in the northern coast (see Atyrau), to 28.5 °F in the east coast (see Aktau), to nearly 32 °F near the border with Turkmenistan, as well as a small southern portion in the central part of the country, near the border with Uzbekistan, where the average is about freezing as well. In general, at a given latitude, the winter temperature decreases proceeding towards the east.
Here are the average temperatures throughout Kazakhstan in January.
January average temperatures in Kazakhstan
The summer temperatures are more uniform, and vary mainly depending on latitude, and of course altitude (but most of the country is flat): in July, it ranges on average from 66 °F in the far north, to 77 °F in the south-central, to nearly 85 °F in the far south.
As mentioned, even the rains vary from north to south: the northern part, the rainiest, is occupied by the Kazakh steppe (or the Kirghiz steppe), while the central and southern regions are semi-desert or desert. Anyway, in the mountains the rainfall increases again.
The majority of Kazakhstan is flat, or in any case it has an altitude lower than 1,600 feet. The only mountainous areas are: in the south and south-east (toward Kyrgyzstan and China), the Tian Shan, which also includes the highest peak of Kazakhstan, Khan Tengri, 22,999 feet high, and in the east (toward China and Mongolia), the Altai Mountains. In mountainous areas, precipitation is more abundant than in the plains, so that forests grow, although it depends on slope exposure (for instance the Charyn Canyon, closed among the mountains, is dry). At high altitudes, above 11,500 feet, there are large glaciers.

Now, let's see in detail some areas and the major cities.
In the west, the area of the Caspian Sea is arid continental in the north coast, with cold winters and hot summers: in Atyrau (former Guryev) the average temperature goes from 21 °F in January to 80.5 °F in July, when days are often torrid. Here are the average temperatures.
Average temperatures - Atyrau
Min (°F)161427435564706654412819
Max (°F)272843637788919077594130

Precipitation is scarce: throughout the year, about 6.5 in of rain or snow fall. Here is the average precipitation.
Average precipitation - Atyrau
Prec. (in)

On the eastern coast of the Caspian Sea, the climate becomes less cold in winter, the January average rising to 28.5 °F in Aktau, and to 32 °F on the border with Turkmenistan, and it becomes also more arid, the annual rainfall being around 6 in.
East of the Caspian Sea, the Ustjurt Plateau is desert, with very little precipitation, about 4 in, cold in winter (in January, the average is around the freezing point or a few degrees below) and hot in summer, with average highs around 95 °F, but with peaks of scorching heat, around 113/117 °F.
To the east, we find the area of the Aral Sea, a former large lake, once fed by the rivers Amu Darya and Syr Darya, but partly dried up in recent decades, due to the exploitation of the same rivers. Here the climate is desert, with around 6 in of rain per year; winters are freezing, with January averages around 14 °F, and summers are hot, with highs around 90/93 °F in July. In Aral (formerly Aralsk), which was once a port before the lake retired, the lowest recorded temperature is -36.5 °F, while the highest is 117 °F.
Average temperatures - Aral
Min (°F)7721415464706452372512
Max (°F)192339647990939077593925

East of the Aral Sea, we find the Baikonur Cosmodrome, which hosted the Soviet space program and has continued to work under the Russian administration; here the climate is similar to that ok Lake Aral, that is, continental and desert.
About 125 miles south-east of the cosmodrome, we find the offshoots of the Kzyl Kum desert, which is very hot in summer. In Turkestan, the maximum of July is normally around 97 °F.
Average temperatures - Turkestan
Min (°F)162132455463666352372719
Max (°F)323754708191979182685036

Astana, the capital of Kazakhstan, is located in the north, therefore in the steppe area, which receives a bit of rain more than the center-south, because it can experience some thunderstorms in summer; precipitation amounts to 12.5 inches per year, with a relative minimum in winter, when it's around 0.6/0.8 inches per month, and a relative maximum in summer, with 2.1 inches in July. Here is the average precipitation.
Average precipitation - Astana
Prec. (in)

Due to its northern location, the temperature in summer is not so high, but it's rather pleasant: the average in July is 68.5 °F. But even here in the north, heat waves in summer are possible, with peaks of 100/105 °F. Summer, aside from the thunderstorms, which in any case are not very frequent, is sunny. Winter is rather cold: the average in January, 1 °F makes it one of the coldest capitals in the world. In the worst moments, the temperature can plunge to -58 °F. During winter, snowfalls are frequent but not abundant, however, the snow covers the ground for long periods. Spring and autumn are very contrasted: in April and October, the temperature can drop below -4 °F, but it can also exceed 77 °F. The clashes of air masses can also cause strong winds.
Average temperatures - Astana
Min (°F)-8-832843525552412812-2
Max (°F)91223486877817566482714

In the far north, Petropavl (formerly Petropavlovsk), has a climate similar to that of Astana, just a bit colder, and a little more rainy in summer: it receives 14 in of rain or snow per year.
Even in the eastern cities of Semey (Semipalatinsk) and Oskemen (Ust-Kamenogorsk), the climate does not differ much from that of of Astana.
Almaty (formerly Alma-Ata), the most populous city and former capital of Kazakhstan, is located in the south-east, in the early slopes of the Tian Shan, and its altitude varies depending on district, however, the center is at 2,600 feet above sea level. Here the average temperature goes from 21 °F in January to 75 °F in July, often with large temperature range between night and day; precipitation is more abundant than in other parts of the country, due to its location at the foot of the mountains, and amounts to 23 inches per year, with a maximum in spring, between March and May, when intense clashes between air masses occur, which is why spring is also a windy season. In Almaty, there are 55 days with snow per year in spite of the southern location, because of the altitude, which exposes it more easily to cold air masses, bringing late snowfalls in spring and early snowfalls in autumn. Here is the average precipitation.
Average precipitation - Almaty
Prec. (in)

Summer is hot, but the altitude makes the heat more bearable than the plains, and it's sunny as well, but occasionally there may be some thunderstorms in the afternoon, especially in the first part of the season (June-July). During winter there can be waves of frost, sometimes prolonged, and in the worst moments the temperature can plunge to -22 °F.
Average temperatures - Almaty
Min (°F)121428435259646152412716
Max (°F)303245637281868473614332

Shymkent is located in the south as well, at 1,600 feet above sea level, and like Almaty it's located at the foot of the mountains, but more to the south-west, in the South Kazakhstan Province, where intrusions of mild air from southern Asia are possible, in fact this is one of the least cold cities in the winter, since the average in January is 27.5 °F, which does not mean that it is not exposed to waves of frost, but only that these alternate with mild periods, while in summer it's hot, with a July average of 79 °F. Rainfall amounts to 21 inches per year, with a peak between November and April: the substantial precipitation make possible heavy snowfalls in winter.
Here is the average precipitation.
Average precipitation - Shymkent
Prec. (in)

Summer is dry and sunny. Even this city is windy, for the clashes between air masses and the proximity of the mountains.
Average temperatures - Shymkent
Min (°F)192336465563666152413427
Max (°F)364154708190959181665441

In the east of the country, north of Almaty, in the area of Lake Balkhash, a large lake made half of freshwater and half of saltwater, located at 1,115 feet above sea level, the climate is strongly continental, with an average in January around 5 °F, and an average in July around 74.5 °F. In a year, just 5.5 inches of rain or snow fall. Here are the average temperatures.
Average temperatures - Balqash
Min (°F)011434485963594632193
Max (°F)121832557281868270543419

When to go

It's hard to find a best time to visit Kazakhstan, because there is not a month which can be free from either cold or heat. In the north and in Astana, in summer from June to August the average temperature is pleasantly warm, but you cannot rule out heat waves with peaks around 105 °F. In the central and southern areas, summer heat waves are more frequent, and the heat is quite often intense, so you may prefer spring and autumn, in particular from mid-April to mid-May, and the second half of September. Around mid-April and mid-September, you can generally find acceptable conditions throughout the country.

What to pack

In winter: for Astana and the north, cold weather clothing, synthetic thermal long underwear, fleece, parka, wind jacket, gloves, warm boots. In the southernmost cities, like Shymkent and Aktau, you can wear lighter clothes during mild days, while it's useful to bring a scarf for the wind.

In summer: for Astana and the north, light clothing, T-shirts, but also long pants, light jacket and sweater for the evening and cooler days; raincoat or umbrella.
For the vast southern plains (Ustjurt Plateau, Turkestan, Shardara), lightweight clothes, loose fitting and made of natural fabric, scarf or desert turban, a sweatshirt for the evening.
For Almaty, Shymkent, Caspian Sea, Baikonur, Aral Sea and Lake Balkash, light clothes, sun hat, scarf, a sweatshirt for the evening.
In the southern mountains (Tian Shan), at intermediate altitudes, light clothes for the day, sun hat, sweatshirt and jacket for the evening, hiking shoes; above 10,000 feet, sweater and warm jacket for the evening; above 13,000 feet, down jacket, hat, gloves, scarf. In the mountains of the north-east (Altai), the cold starts at lower altitudes: at 5,000 feet you already need clothes for spring and autumn, like sweater and jacket.
For women, it is best to avoid shorts and miniskirts, especially outside the big cities.