Buses in Almaty are big city buses, not cramped marshrutkas like in Bishkek. They are very full in peak times though, from 8-10am and 5-7pm on weekdays. Buses drive from 6am to 9pm, except for 1 night route from the airport to the city center.
The price is 80 tenge for a ride. You can take your bicycle on the bus, but it will cost you double.
To pay for the bus ride, you now need the Onay card, sold at kiosks or metro stations for 400 tenge. You might need to search to find a kiosk that sells them, they seem to run out quickly. You then need to charge the card at a payment terminal, found in kiosks or shops. You get 1 free ride for your 400 tenge and you can sell the card back at the train station or at the head office of Onay at Ryskulbekova 33/1. It’s seriously far away though.
Be aware that it takes 15 minutes for your card to recharge, and that you cannot beep in someone else on the same card if you are 2 on the bus. The card is only valid in Almaty and can also be used for the metro.
Since the Onay card is new and still a bit of a hassle (and because people are used to doing things their way), lots of people still just pay the bus driver. There are ticket inspectors, but they don’t give fines; if someone hasn’t paid, they tell them to pay, or in extreme cases, to get off the bus.
2 great sites/apps have everything you need to know about which bus routes go where in Almaty. Citybus.kz is the best. You don’t need to know Russian to understand how it works, just click your point of departure and destination on the map and it will show you which bus to take. More extensive guidelines are posted here. The app is Android-only at the moment.
Zaproezd is the other one, in case Citybus does not strike your fancy.
Have a nice trip!