As I've discussed before in other countries' posts, a marshrutka is kind of half way between a van and a minibus. They're pretty common in former Soviet countries, and even the name is Russian. For Kyrgyzstan though, this is the main form of public transportation for the entire country. Here there are no metros, not even in the capital Bishkek, and trains are few and far between. Thus leaving these odd vehicles as the sole option in getting from point A to point B.
They go pretty much everywhere, doing loops and routs both within cities and across the country, like the six hour one I took from Bishkek to Karakol. There are even a few that go internationally to neighboring countries like Kazakhstan and Tajikistan. However, before I praise them too much, I do wish to note that they are usually jam packed as tight as possible (since that's how the drivers make their money) and ther is usually no air conditioning. If you happen to be in one for over 15 minutes, there will be sweat.
But that said, they are dirt cheap, usually costing just 10 som (about 15 cents) to ride one within the city. Even the one going across the whole country cost me just 350 som ($5), so wherever you go, you'll never have to worry about the price. Just prepare yourself though if the ride is long since most of the roads are not paved and marshurtkas have no seatbelts. You will be bouncing up and down a lot.
Usually I hate taking taxis anywhere. They're so often over priced and they take all the fun and adventure out of going from place to place. But in Kyrgyzstan, especially if you're going somewhere that isn't on a Marshrutka route, they're the only option you've got. This comes with some pluses and minuses, as you can probably imagine.
On the plus side, taxis in Kyrgyzstan are ridiculously cheap compared to other countries. Granted they are more expensive than marshrutkas, but you can still take a 20 minute ride in one for just about 100 som ($1.40 or 1.30 euro). Especially if you happening to be splitting it with others, you'll reach your destination significantly faster than by marshrutka (which will make several stops and pick up other passengers) and you won't have to worry about the cost.
That said, taxi driver will often try to scam and overcharge you, so be prepared. Make sure you agree on a price before getting in a cab and always haggle it down. Also, be prepared because these drivers go outrageously fast and most cabs do not have seat-belts.
To say hitchhiking in Kyrgyzstan is easy would be a huge understatement. For example, activities that are more difficult include tying your shoes, washing your hair, and clapping your hands. I kid you not, people stopped and offered me a ride just upon seeing me standing near the road. I didn't have to stick out my thumb or even look up. If I were to break it down to statistics, I would say about one out of every three drivers will stop for you and one of three that stop will be going in your direction. Waiting more that five minutes is rare, and waiting more than ten is unheard of.
If you're not in a huge hurry, definitely choose hitchhiking over a taxi any chance you get. It's free, more fun, and actually quite a bit safer since taxi drivers go like maniacs. And on top of that, you can actually get a pretty good conversation out of it since the driver will likely want to get to know you a little. Kyrgyzstan doesn't get many tourists, so people are usually pretty excited to come across foreigners.