Use 'swimmingly' when you want to express that things are proceeding better than expected and with positive results. Her new business is going swimmingly, exceeding all forecasts.
Pro Tip 2/3
Informal Contexts
'Swimmingly' is casual and often used in social situations or personal updates, not in very formal writing. I heard your piano lessons are going swimmingly!
Pro Tip 3/3
Figurative Usage
Despite involving 'swimming,' it's not literal and doesn't relate to water, but to smooth progress. The software installation went swimmingly, with no issues.