Kaunas Lithuania

What do you mean by safest? An unusual question. A taxi or a hired driver is the most convenient. Are you concerned about accidents or theft or what?

There is an inexpensive shuttle train that connects the Vilnius airport with the main train station. It takes about 8 minutes. From there you could take a train (http://www.litrail.lt) to Kaunas or walk across the street to the bus station and take the bus (http://www.autobusubilietai.lt/Popular-bus-routes-in-Lithuania/Vilnius-Kaunas-stotis/). Both of these modes of travel are safe. Though I don't know if cost is an issue, a hired taxi would be quite expensive. Good luck.