By scheduled bus, the route from both cities is to Mostar or Čapljina and then a local connection or taxi for the last stretch, so allow a half day; direct coaches to Medjugorje run on some days in season. Timetables change by season and are not published by the parish or the State Department, so check the bus station sites in Split and Dubrovnik for the day you travel rather than relying on a general figure. Pre-booked shuttle vans and private transfers, arranged by your guesthouse, are the usual choice for groups of three or more and go door to door without a change.
Taxis from either airport will cross the border, but agree the price before leaving and expect the driver's documents to be checked along with yours. If you rent a car, tell the rental company you are taking it into Bosnia and Herzegovina: a cross-border authorization and a green card insurance certificate are required, and drivers need an international driving permit with their U.S. license; snow tires are mandatory in Bosnia from November through April.
At the border, have passports ready for two sets of checks, Croatian exit and Bosnian entry; queues are longest on summer weekends and on big feast days such as June 25 and the Youth Festival in early August. U.S. citizens need no visa for up to 90 days. The existing page on whether to fly to Split or Dubrovnik compares the two airports, and the one on day trips covers Mostar.