She takes not only swimming and tennis but also singing and dancing classes.
Literal
She [topic-は] swimming [and-や] tennis [possessive-の] classes [needless to say-はいうまでもなく] singing [and-や] dance [possessive-の] classes [also-も] is-taking.
~はいうまでもなく means 'needless to say' or 'not to mention' — it establishes one thing as obvious before adding something extra. The structure says: 'Swimming and tennis go without saying, but she also takes singing and dance classes.' This pattern layers expectations: what you'd already expect, plus what's surprising.