In addition to the great stuff @heresjohnny suggested, I’d recommend going to the Reichstag—it’s like our capitol building, and is right next to the Brandenburg Gate. Those are both at the subway stop Unter den Linden. Then if you just keep walking a few blocks east on the street Unter den Linden, you’ll get to Museum Island…which, as you can imagine, has a few pretty sweet museums. Oh! And on the way to Museum Island, you’ll cross Friedrichstraße, a street famous for having lots of ritzy stores, if you’re into that.
The cool church that got its roof blown off in WWII and they decided to leave as a memorial is right outside the Zoologisher Garten stop. (The subway system there is great, by the way.) The street Kurfürstendamm, which runs right along that plaza is also pretty famous. And the super tall TV tower, you can go up is at Alexanderplatz, which is also where there’s this world clock…which is famous for some reason I don’t actually know.
Oh, and you definitely have to check out some Christmas markets. They’re super German and just really fun. They have quite a few in Berlin, probably one for every district. Last time I was there (three years ago), there was one across the street from Museum Island, if you go there. And there’s definitely going to be one by that memorial church at Zoologischer Garten.
As far as running goes, I also think Tiergarten is a safe bet. I also remember biking the trail they have part of the way along where the Wall ran…but I’m afraid I have no idea how to access it.
Oh man, I wish I could go back! I hope you have a blast!