They can be omited in speech when it's clear from the context to whom you refer. In written text, they are omited in a colloquial style that closely follows the speech (for example, in a chat conversation), or in a written text that describes the same person (e.g. in a biography, when using pronouns would require using the same pronoun over and over).
The phrase «Не хочу́ говори́ть о му́зыке» is colloquial. You wouldn't use it in a formal document.
In general, using pronouns is always correct, dropping them is sometimes allowed but it's not always correct. So, to be on the safe side, you could just use the pronouns everywhere.
Addition. I've missed an important point. In written Russian, you can also omit the pronoun in subordinate clauses:
- Она́ зна́ет, что [она́] винова́та. 'She knows she is to blame.'
- И́горь сказа́л, что [он] прие́дет. 'Igor said he will come.'
- Тогда́ Окса́на ещё не зна́ла, что [она́] ста́нет изве́стной худо́жницей. 'Back then Oksana didn't know that she will become a famous painter.'
So this sentence kind of works the way the exact English translation would. In English, it’s okay to say “don’t want to talk about music” in informal speech.
They're different forms of the same word used depending on the following sound (like "a" vs. "an" in English):
о - used before most consonants
об - used before vowels
обо - used before certain complex consonant clusters like мн, чт, вс
o is used before words that begin with a consonant sound, which includes vowels like e.
об is used before words that begin with a vowel sound
обо, before the consonant clusters as set out by dirckk