October in Germany can still be mild and sunny, it is then referred to as a Goldener Oktober.
The leaves are turning yellow, red, orange and brown. It is the perfect time to take beautiful nature photos.
Traditionally there should be many harvest and wine festivals, but due to the Corona pandemic, most have been cancelled this year.
If you are interested in wine, take a drive on the German wine route, which is the oldest scenic drive in Germany. You can find more information here.
A boat ride along one of the large rivers, such as the Rhine or the Neckar, will reward you with spectacular autumn vineyard views.
It is also a good time to go hiking, just make sure you have the correct clothing and shoes.
If it is getting too cold outside to do sport, try some of the indoor sports, such as indoor soccer, quash, badminton (das Federballspiel), volleyball, handball or ice skating.