Public transportation in Almaty

I am lucky.  I live on the trolley line.  There are so many busses and very few maps.  My advice would be to get on a bus and see where it goes.  Just have some tenge in your pocket in case you get hopelessly lost.  A gypsy taxi driver will take you any where.

The public transport system in Almaty is actually alright, despite what many local Kazakhs have told me.  You can take the bus, trolley bus or tram almost anywhere in the city if you know what number to take.  I myself only really use the buses, they cost 80 Tenge regardless of where you are going and you will either pay a conductor, the bus driver or a machine on the bus.  Buses are usually quite busy and you will find yourself in cramped situations often, although there are times when they are empty.  Also if you use buses you may want to look out for the hierarchy of who can sit.  Generally if you are young there's no point sitting down on the bus.

There is also the underground, which is pretty limited at the moment with perhaps six stations, but it is set to be extended in coming years.  I don't know the price as I am yet to use it.

There is also travel by taxi, which can mean flagging down any old car (sort of like thumbing a ride, except that you pay).  This method of travel is more expensive than proper public transport, but its effective and relatively cheap.  You need to know the street names of where you want to go for this and also need to know some numbers in Russian, to state your price and/or to barter with the driver.  This method of transport, from my experience is safe, but at the same time one should be careful, since you are getting into someone else's car.

Or there are proper taxis, these can be flagged down as well, but more likely you will call one, they know the roads better and can take you almost directly to where you want to go, but you will be looking to pay 2-3 times more than just getting a random car.

Anyway that's my evaluation of Almaty transport, but I have only been here for a short time and I am sure others can add more.

I agree with Giras, before jumping into any one of the mentioned option, learn some russian. it will help you bargain and convey a message to the driver that you are not a newbie and definitely not someone to fool around with.

I have seen many girls opting for cars/taxis instead of bus and that can only happen if its safe. So thumbsup from me too.

And yes call cabs costs more, but then they are at your mercy. use them the way you like.

Oh, actually, can anybody tell me at what time do the buses stop running at night?

The price for all public transport is 80 tenge now. They start generally at 6 a.m (some of them even earlier i think) and officially should work till 10-11p.m., but in fact most of them finish at 8-9 p.m.
About taxi services- now here are some of the companies that provide nice prices, but not cheaper than 300-400 tg, for the shortest distance. Some of them offer 500 tg in the square (Dostyk-al Farabi-Sain-Raimbek), about 2000 tg to the airoport. But you should book it in advance.