A Bard knows six spells at 1st level; two cantrips and four 1st-level spells. From there, a Bard learns one new spell per level, except at levels 10, 14, 18, 19, and 20.

At levels 10, 14, and 18, a Bard gets access to magical secrets, which allows them to learn two spells from any class.

At levels 19 and 20, a Bard does not learn any new spells. So, at level 18, a Bard knows 22 leveled spells, the maximum it will ever know.

At levels 4 and 10, a Bard also gets access to one new cantrip, totaling 4 known cantrips. So, in total, a level 18 Bard knows 26 spells in total; 22 leveled spells and 4 cantrips.

Bard Cantrips + Spells Known by Level

Player’s Handbook, pg. 53