If something goes swimmingly, it means it goes very well and without any problems.
When something goes swimmingly, it feels easy and happy.
Examples:
The party went swimmingly, everyone had fun.
The project is going swimmingly, we're almost done.
The game went swimmingly, we won by a lot.
The picnic went swimmingly until it started raining.
She thought the test would go swimmingly, but it was really hard.
The plan went swimmingly, and we all got ice cream afterwards.