Also a riverboat trip on the Spree, either in the city centre or a longer trip from Treptow to Kopenik and the Muggelsee lake. Kopenik is a charming historic town on the eastern edge of the city and the Muggelsee is a large lake with wooded shores, where the boat will pull up at a large beer...
Get the five day travelcard (sorry i've forgotten the actual name for it) which covers all local trains (S-Bahn) underground (U-Bahn) and buses, very good value. I was there two years ago and encountered no beggars on public transport. A colleague was there two months ago and had no such issues...