The exact relation between Dacians and Getae is unclear; ancient sources tell us that they spoke the same language, and some of them claim that they were the same tribe under different names. They should thus be considered related, possibly identical. Wikipedia has a review of the sources.
As for the connection to Goths, it was a common identification in late antiquity, probably based on the closeness of the names. However, the names are most likely entirely unrelated: "Getae" is probably greek in origin, "Goth" is Germanic. The Goths, when they first appear in recorded history, lived north of the Black sea and later expanded towards the south and west, including the former Dacia. (Much has beeen written about the Goths coming from southern Sweden, and later Poland, but this is based on late recorded myths, etymological speculation, and somewhat questionable identification of material culture with tribal identity.)