I am sorry to say that public transportation in KY is virtually non-existent. It is truly a shame. We do have nice new buses/trolleys for transport within Frankfort. You can take Amtrak from Indianapolis or Columbus to Cincinnati and then take a a Greyhound bus from Cincinnati to Lexington KY. Or you can take Greyhound all the way from Indianapolis to Lexington. Greyhound gives passengers traveling together a "companion fare" discount so be sure to click that if buying online or ask. The trickiest part will be getting to Frankfort from Lexington. If you stay at a Bed and Breakfast in Frankfort the owners may be willing to provide transportation (probably for a fee but you never know maybe just gas cost). Frankfort cab companies Bluegrass Cab (502) 682-3393 and Capital City Cab (502) 875-5078 offer service in the city 24 hours a day. I've been told they cost less than the Lexington cabs but you may want to compare. I checked with two hotels- Capitol Plaza and Fairfield Inn and they do not have a shuttle. I will be happy to try to answer any other questions you have re: Frankfort and KY.
Commonwealth Cab has proven to be the most reliable in Frankfort. Friendly drivers, prompt service, and the most reasonably priced. 502-803-8294
Not sure if you have made your trip or not, but Miller Transportation has bus service from Lexington to Frankfort. The link is www.millertransportation.com/services.html. Click on the "Regular Route Service" n the bottom of the left-hand column for pricing and times.
It looks like it's roughly a $27 ticket roundtrip, though I believe it depends on which days you travel.
Hope this helps.