Kutaisi airport in Georgia is a new airport, with relatively few flights at present. However, due to Wizzair's direct flight from Kiev, it is now the cheapest location to arrive in if you are coming to the Caucasus from Europe. It is not easy to find out information about local transport to and from the airport, so I hereby place this information on the internet, hopefully to help other people.
The simplest way into Kutaisi is simply to get a taxi. A fare of about 25 Georgian Lari (£10, $15) seems to be the going rate. Whether locals get a cheaper price and/or taxi drivers will try to get away with charging you more if they can remains to be seen. On the other hand, the distance is about 30km, so this isn't particularly unreasonable.
When I arrived on January 28, there were two public minibuses (marshrutkas) waiting outside the terminal, one going to Batumi (2 hours away) and the other going to Tbilisi (four hours away). I don't know if such services go after every flight, but they are likely useful. Of course, they also likely go in the other direction, but when and where they depart from in Tbilisi and Batumi is not information I was able to find out, even with the help of a Georgian in Tbilisi. (as the Tbilisi bus goes in the direction of Kutaisi, it is likely possible to ask the driver to take you to Kutaisi for a few Lari, but I suspect that this depends on how full the bus is).
However, the key piece of information is this.
Kutaisi airport is on the main highway in Georgia (Georgian Highway number 1, European route E60), in the sense that you walk out of the terminal and past the car park, and the exit to the carpark turns straight onto the highway. Locally, this is the road from Kutaisi to Samtredia. (The airport is actually closer to Samtredia). To get into Kutaisi, you should stay on the same side of the highway as the airport terminal, and flag down the next marshrutka going in that direction.To get from Kutaisi to the airport, you should get the minibus to Samtredia, and ask the driver to let you off at the airport.
This should be simple, once you know this. (To get from the airport to Samtredia, cross the road and flag down a marshrutka going in the opposite direction. There is a bus shelter at which to wait).
Similarly, to get to Kutaisi airport from Tbilisi, you should get a bus to Samtredia and ask the driver to let you off at Kutaisi airport. (Getting on the bus to Poti likely works, too). Coming from the coast, get on the bus to Kutaisi, and ask the driver to let you off at the airport. In a pinch, get off at Samtredia, as this is much closer to the airport and going all the way to Kutaisi will involve 30km of backtracking.
When I was there in January 2013, there were no money changing facilities or ATMs at the airport. As the airport was brand new and still being finished at the time, this might be rectified soon. If not, this is unlikely to be a problem if the service you are boarding terminates at wherever you are going, as there are many money changers and ATMs in all Georgian towns, and getting a little money to pay the driver at the end of the route is feasible. If you are getting off in the middle of a route, this might be a little more problematic, and there is something to be said for having a few small denomination euros or dollars handy. (Good advice anywhere). If none of this works, appealing to the kindness of strangers is also likely to work. Georgians are very warm and hospitable people to visitors, and are likely to be sympathetic.
Update (19/2/13): I see that Wizzair have just announced that they are increasing the Kiev route to daily, as well as adding flights from Kutaisi to Warsaw, Donetsk and Kharkiv. That Warsaw flight is going to make it even easier for those of us trying to get to the Caucasus from Europe. Hopefully, the increased traffic will also encourage the owners of the airport to, say, install an ATM.