Is it possible for a Bard to learn Occult spells without using the Skill Occultism? Is there any other method. Or is it a Skill you must increase? I have three other skills I want to increase instead.
Yes, spellcasting (but not rituals) is not impacted by whether you have a certain proficiency level in the associated skill. The Bard does, however, have some feat options that are improved by having legendary in occultism.
You only need to use occultism's Learn a Spell if you have the bard feat for a spellbook or want to learn an uncommon spell. And you can also do those with arcana if you take Unified Theory at level 15.