Learn Finnish: Puhelimen väri on musta. - The phone's color is black.

It is in the genitive case to show possession, meaning "the phone's". Words ending in -in, like puhelin, change their stem to end in -ime- before adding the genitive -n. That is why it becomes puhelimen.

Could I just say puhelin väri instead?

No, Finnish requires the genitive case to link these nouns, similar to the English 's. Just putting two basic nouns together without the genitive -n would sound incomplete, so you need puhelimen to specifically mean "the phone's color".

I often see colors in the partitive case, like mustaa. Why is it just musta here?

You use the partitive (like mustaa) when the subject is an uncountable mass noun, such as in Kahvi on mustaa (Coffee is black). In this sentence, the subject is väri (color), which acts as a standard, countable singular noun. Because of this, the adjective describing it stays in the basic nominative form musta.

I have heard people use musta to mean "in my opinion". Is that related?

It looks identical, but it is just a coincidence! The basic adjective for "black" is musta. However, in spoken Finnish, musta is also a very common shortened form of minusta (from me / in my opinion). In this context, it simply means black.