Tourists renting a private car and driving in Kyrgyzstan is virtually unheard of and not recommended. The roads are in poor shape, police are highly corrupt, auto insurance doesn't exist, and hiring a taxi is too easy and cheap to make this an option. Long-term foreign residents frequently drive, but many opt to use a driver.
Buses And Taxis
Minibuses marshrutkas and shared taxis are the most common and accessible option for traveling within Kyrgyzstan. They're amazingly inexpensive and congregate at every village center or bus station. You can also arrange a private taxi by purchasing all the seats at the bus station or contacting a taxi firm directly.
The prices for mini-buses are set and straight-forward, but it won't generally leave until it is full and you may find yourself holding a child in your lap. With shared taxis you will be quoted a price for one seat and if you have significant luggage you should expect to pay for an extra or partial seat. You should negotiate prices, but as a foreigner you will likely pay more than a local.
There are several daily flights between Bishkek and Osh. There are also a few flights a week between Bishkek and Jalalabad and Batken. The flights are operated on local airlines using 30-40 year old soviet planes. On the other hand, the mechanics and pilots are well trained how to operate these old beasts.
The only domestic rail link is between Balykchy Western edge of Issyk Kul to Tokmok through to Bishkek through to Karabalta and on to the Kazakh border. The trains take at least twice as long as a taxi, but are half the price and you get to meet a lot of interesting folks, mostly pensioners, that need the 40-80 soms they would save by taking a mini-bus or taxi.
The concept of free rides is not really understood here. Particularly if you happen to be a foreigner. Most drivers will expect you to pay a small sum of money for gas. Either you can try to explain that you do not want to pay, the russian phrase Bez deneg can be used. Alternatively you can just pay the sum.
If the driver is asking for too much money you can always haggle! As a rule of thumb you should either pay the same price you'd pay for the bus or lower.
This is the real way to see Kyrgyzstan by the saddle of a horse. There are several tourist agencies that can make it happen for you, as the Kyrgyz are famous horsemen dating back to the days of Ghengis Khan. It is said that all Kyrgyz are born on a horse, although with growing urbanization that seems to be less common.