I am planning a trip from Berlin to the Harz, specifically the peak Brocken. As finding information turned out to be quite hard, I decided to sum things up here in a short article.
The train ride
Two options: ride a Deutsche Bahn train, any day of the week, almost any hour. Or ride the famous Harz-Berlin-Express (HBZ) operated by Abellio, but only on Saturdays and Sundays. Both types of trains can be booked on the DB website. You can also book the HBZ tickets directly on Abellio's website.
With a bike
Brocken is located in the german state called "Saxony-Anhalt", where taking a bike on a train is free (information found here and here). Therefore, when bringing a bike on a Deutsche Bahn train, just book a bike ticket only for the part Berlin-Genthin. In other words, in the DB Navigator mobile app, just get a VBB Fahrrad ticket for the overall network of VBB (3.6 euros).
If riding a Harz-Berlin Express train, you must book a 3-euro bike ticket. I could not find where to book them online, so do it at a train station.