There are no direct trains from Almaty to Bishkek. You can only go to Bishkek by bus, shared taxi or airplane.
Almaty – Bishkek by bus
There are no buses going to Bishkek, only minivans (commonly called marshrutkas). Minivans to Bishkek leave from Sairan bus station when they are full, which doesn’t take long. The price is 1200 tenge. It takes 4 hours to get to the border.
After the border, it might take a long time for your bus to come through. You can already leave with a local bus if you have some small Kyrgyz money. They will drop you off in the center of town, whereas the bus will drop you off at the bus station at the edge of Bishkek.
Almaty – Bishkek by shared taxi
Outside of the Sairan bus station, shared taxis await. A good deal is around 2500 tenge for a seat in a shared taxi to Bishkek. Most of them will not cross the border, so you will have to do the last bit with local transport.
Almaty – Bishkek by airplane
A ticket Almaty – Bishkek from Air Astana costs 80$. Flights are a few times a week. It takes 1 hour to get there.