Short answers, pulled from the story.
The murder of Amalasuntha in early 535 provided Emperor Justinian I with the pretext to launch the war. Justinian declared war after his cousin Theodahad arrested and killed Amalasuntha, who had served as regent for her grandson Athalaric.
Belisarius entered Rome unopposed in December 536. He later secured Ravenna in May 540 but refused to sign a treaty that would partition Italy.
Totila seized the throne in 541 and reoccupied Rome in 546. He tore down one-third of the city walls before leaving to pursue the retreating imperial forces.
The siege ended when part of the Byzantine garrison opened the Porta Ostiensis gate for Totila. His men swept through the city, killing all but the women who were spared on Totila's orders.
The Battle of Taginae took place in 552 when the Byzantine general Narses faced Totila. Totila was killed in the rout, and the Gothic army broke and fled.
The Gothic War ended in 554 with a pragmatic sanction that prescribed Italy's new government. Italy was devastated and depopulated, with great cities abandoned and urban society destroyed.