City- airport transfers

If you have an InterRAIL Pass you can use in Berlin the S-Bahn (train that stops at every station) or the FEX (Express Rail Service), about taxis I don’t know. 

In Prague source interrail.eu:

When flying into Prague, you'll arrive at Vaclav Havel Airport Prague. The airport is quite close to the city centre of Prague - it takes around 30 minutes by bus. The Airport Express (AE) buses depart in front of terminal 1 and drop you off at Prague central train station (Praha Hlavni Nadrazi). Tickets cost 60 CZK (Czech Koruna) for adults and half for children

I'm not sure what's wrong with the train from Berlin airport, especially if you want to get to Berlin Hbf, but airports usually have information pages on how they're connected, see Berlin airport and Prague airport, with more options.

Just as in the UK about any major and most minor airprots do have links-mostly bus but the bigger ones now often also a metro, tram or train. A real train (=not a metro!) is included in the pass. There is even a special sites al dedicated to that-google will also help with these basic questions. Its like gettingtoandfromairports.

In Praha you probably stay before-the local excellent transit is run as DPP and offers still quite cheap tickets, also for 24/48 etc, hrs. Its famous for its many red trams. Students with ISIC and <26 may even get reduced fares-check their site. DPP is free for seniors >65 (from EU ONLY!) but you have to get a fotocard for that first. In fact on INland trains/buses in CZ the discount for those is 75%, making it far cheaper as passdays.

Most people make these tickets valid untill departure-they are not valid on the expensive AE, but the local bus from the Dejvicka metrio=just follow crowds and signs. Single tickets MUST be datestamped before use!