After some further searching I found the program screen or rather the command how I can connect to my port.
screen <serial_port> <baud_rate>
screen /dev/ttyACM0 9600
Then were my data shown and I could save those with the key combination 'Ctrl-A' and then 'H' into a file ( log-file ).
If I understood correctly there also is the possibility to save all the information which was shown until the moment you type the following commands into a file:
You have to be inside screen and then type 'Ctrl-A' and then ' : ' and then you have to type 'hardcode -h ' where the name of the file is in which everything will be saved.
I also tried PuTTY but couldn't connect to my port. I have to say that I tried it just two times because at that time screen already worked.
Maybe someone can help me with PuTTY. I would like to know how it works with it, too.