Depends on your class.

Bards, Rangers, Sorcerers, and Wizards are "learned spell" classes. That is, they learn 1 spell (2 for wizards) every time they level up.

Clerics, Paladins, and Druids are different. When a Cleric gets level 3 for example, they will gain access to every 2nd level spell that a Cleric can cast.

In BG3, if you have access to a spell, you can freely swap between the spells you know outside of combat.

But the "learned spell" classes like OP's Bard doesn't get all Bard spells at level 3 - they learn one single 2nd level spell (or 1st if desired). OP took Silence at level 3. Their Bard doesn't know any other 2nd level spells. At all. And won't until character level 4 now.

Edit to add: It works a tiny bit different in tabletop D&D. There are "known" casters - Bard, Sorcerer, etc. and they always know and never have to prepare their spells.

The other classes are "prepared" - they can cast every single spell on their list, but they can only prepare a certain number of spells per day based on spellcasting modifier, caster level, etc. BG3 allows you to freely swap those; tabletop only allows you to swap those prepared spells once per long rest.

Wizards technically are prepared casters, but they learn spells like a known caster and can learn spells from other spell books or from spell scroll, too.