Efter at vi havde hentet dem skulle vi køre til Rostock i Tyskland. På vej derhen stoppede vi et par gange for at gå på toilet eller købe noget at spise.
After we had picked them up we had to drive to Rostock in Germany. On the way there we stopped a few times to go to the toilet or buy something to eat.
I have made a minor change because your original wording (After having picked them up) is 'double' in English. There are several ways to say this and you picked one which is not gramatically correct
Some options:
After we (had) picked them up, ...
Having picked them up, ...
After picking them up, ...