Do you have any advice on my upcoming trip to Berlin?

I’ll be visiting my best friend, who moved to Berlin a few months ago, from December 14 to 28. We don’t have much of an itinerary planned yet, so are there any sights we must see? Places we have to eat, or things we have to do?

For bonus points, if anyone knows the city particularly well, can you give me some suggestions on how I can keep up running while I’m there? I imagine the streets might be snowy/icy, something I don’t have to worry about in Southern California. Are there trails that might be well-maintained, or an indoor track I could have access to temporarily?