We study ~330 massive (M > 10M), newborn spheroidal galaxies (SGs) around the epoch of peak star formation (1 <z <3) to explore the high-redshift origin of SGs and gain insight into when and how the old stellar populations that dominate today's Universe formed. The sample is drawn from the Hubble Space Telescope (HST)/WFC3 Early-Release Science programme, which provides deep 10-filter (0.2-1.7 μm) HST imaging over one-third of the GOODS-South field. We find that the star formation episodes that built our SGs likely peaked in the redshift range 2 <z <5 (with a median of z ~ 3) and have decay time-scales shorter than ~1.5Gyr. Starburst time-scales and ages show no trend with stellar mass in the range 10 <M <10 M. However, the time-scales show increased scatter towards lower values ( 10M, and an age trend becomes evident in this mass regime: SGs with M > 10M are ~2 Gyr older than their counterparts with M <10M. Nevertheless, a smooth downsizing trend with galaxy mass is not observed, and the large scatter in starburst ages indicates that SGs are not a particularly coeval population. Around half of the blue SGs appear not to drive their star formation via major mergers, and those that have experienced a recent major merger show only modest enhancements (~40 per cent) in their specific star formation rates. Our empirical study indicates that processes other than major mergers (e.g. violent disc instability driven by cold streams and/or minor mergers) likely play a dominant role in building SGs, and creating a significant fraction of the old stellar populations that dominate today's Universe.