The public buses cost KZT90. Get on the bus and a conductor will come and ask for your fare and give you a ticket. Given the simplicity of the process, you don’t need to speak the language to be able to do this. The buses are efficient and comfortable, but they are often full. Some buses are from the 90’s, but there are also newer buses from the last 10 years that match any modern city in terms of the quality of the bus.
With the online bus routes map, you can either select the bus route in the top left, or you can click the ruler in the top left of the map, then the first double-click will represent departure and the second double-click will represent destination. You can even move the markers. Buses (or combinations) will appear in the left column. Once you’re on the bus, there is a route map, so it’s worth taking a photo of that on your phone for later trips around the city! When you get on a bus tell the conductor your destination and they will either tell you when to get off or gesture you to get on another bus.
The number 12 bus does a nice loop around the new part of the city with the palace etc., so this is a nice one for a tour. There is also a red double-decker Astana tour bus that is get-on-get off. You can usually find it at places like Baiterek when the weather is warmer.