Being able to travel alone is a great privilege of the program, although the travel rules depend on what organization you are with. I was recently able to visit one of my friends in Stuttgart as my first overnight trip! I had an amazing time being able to see one of my friends from camp and see a totally different part of Germany from where I live.
The process is fairly easy with Experiment e.V., you just need to fill out a form of where you are going and have both your host and natural parents sign it! I felt that the most complicated part was my travel, both the booking, and execution. I did end up flying to Stuttgart as the flight was as cheaper than the train and then took the train back as the airline does not operate on Sundays. Trains are always a little hard with the delays and platforms, but are totally manageable and easily masterable in your time here!
Showing up to a different host family with my friend was a little weird at first, with the different accents and a totally different dynamic. I just had to concentrate more on what they were saying and ask them to speak a little slower a few times. Big cities tend to speak more clear German than the dialect of the region, which I think helped! The family dynamic was totally different than mine, which was perfectly fine, I would just say to always keep an open mind when going to another host family in order to enjoy your experience and respect the other host family!