Dam Square is an easy next step from the old centre: the Royal Palace opens to visitors when it is not in royal use, while De Nieuwe Kerk hosts exhibitions and organ recitals rather than regular services. Check the palace calendar before setting out, and keep the visit flexible-both work better as one focused stop than as a forced museum marathon.
For families with older children, the Jewish Cultural Quarter brings the Portuguese Synagogue, Jewish Museum, National Holocaust Museum and Hollandsche Schouwburg into one area. The Holocaust Museum and Hollandsche Schouwburg are recommended for ages 10 and up, so I'd pick the right venue for your family rather than trying to cover everything.