I visited Coburg by taking a direct, regional train from Nurenberg (it takes about 2 hours and 15 minutes). I purchased a "Bayern" ticket at the ticket window and this seemed cheaper than even the internet prices I saw (15 Euros per person).
HOWEVER, if travelling to and from Berlin, there is ICE train service (****pls. see disclaimer below). Until there is direct Berlin-Coburg ICE service (in about 2018) this is a very comfortable way to travel: take the ICE to or from Lichtenfels (about 80 Euros). Lichtenfels is only 20-25 minutes by taxi (about 35 Euros) from Coburg and has much more frequent, direct service.
****DISCLAIMER: the ICE trains are very modern and have very good drinking and eating facilities. However, they have had some disturbing safety issues over the years which you can read about on Wikipedia. So, take your chances as I did- all worked out very well in terms of travel time.
Coburg is almost exactly half-way between two international airports, with Erfurt to the north and Nuremberg to the south. Nuremberg is better served, usually has cheaper flights, and is easier and quicker to get to on the train. Erfurt, however, is served by Ryan Air, meaning very cheap flights to the UK, from as little as one euro plus tax to London, Stansted. Both airports are served by Air Berlin, another budget airline, but Nuremberg gets most of the best deals. Nuremberg is also served by a number of other airlines, meaning destinations as far afield as Kiev. Nuremberg is particularly convenient for cheap flights to Budapest, which can be had for as little as 68 euros return, including taxes.
Nuremberg International Destinations
Coburg has a station, but it isn't on any of the main lines, and so it is a pain to get to anywhere. Lichtenfels is the closest rail nexus, and is on the main ICE line from Munich to Berlin. If you want to go east to west, to Frankfurt or Prague, you will need to go all the way to Nuremburg, which is a good hour and a half away. Trying to use the regional trains to go to the north to the cities of former East Germany, like Weimar and Erfurt can be difficult and time consuming. If you want to go anywhere by plane or bus, you will likely need to go to Nuremburg as well.