One of my favourite things to do in Berlin is to hop on a bike and explore. Whether it's a forest on the fringe of the city, the graffiti-embellished streets or one of the many urban parks; Berlin is an excellent city to explore by bike. If you are planning on visiting, I strongly encourage you to forget about public transport and cycle your way around (after all, it's what the locals do).
I could spend hours in Treptower park (in fact, I did whilst we were there): sunbathing on the jetty that juts out from Insel der Jugend, lazing on a pedalo and cycling through the woods that surround Spree Park.
I've always wanted to brave the guard dogs and hop the fence at Spree Park (google it - Berlin's abandoned theme park). On our second day in Berlin, we spent hours photographing what we could see through the fences and overgrown vegetation (and a very brief hop over the fence by Abigail), before the distant barking of dogs scared us off). Later on in the holiday, my wish came true and we managed to book ourselves on an elusive tour of the whole park!