Learn Finnish: Me pidämme musiikista ja tanssista.

This is because of verb rection. The verb pitää (to like) demands that the thing you like is in the elative case, which means adding the -sta or -stä ending. Because you like both things, the rule applies to both words.

Why does musiikki lose a 'k' when it becomes musiikista?

This is a reminder of consonant gradation. The base word is musiikki. When you add the -sta ending, it closes the syllable. This forces the strong double consonant kk to weaken into a single k.

Do I really have to add -sta to every single thing I list?

Yes. In Finnish, when you use ja (and) to connect words that are the object of the same verb, they all must take the same case ending. The verb pidämme applies to both words, so both need the elative ending.

Could we use the verb tykätä in this sentence instead?

Absolutely! You could say Me tykkäämme musiikista ja tanssista. The meaning is exactly the same, and both pitää and tykätä require the -sta or -stä ending. The verb tykätä is just a bit more common in casual, spoken Finnish.