All in all, how many spells should I start with as a bard in 5e? I know how many slots, but how many should I actually know and choose from?
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You know the number of spells listed in the Spells Known column of the level progression table for the Bard class, the same table that lists the number of spell slots you have. This table also includes a Cantrips Known column to indicate the number of those that you. As an example, at Level 1, you know 2 cantrips and 4 spells. This information is very clearly indicated in the table. Each class has a corresponding table which appears on the first page or close to the first page of the section detailing the class.
From PHB p.52-53, "Spellcasting":
You know two cantrips of your choice from the bard spell list. ("Cantrips")
You know four 1st-level spells of your choice from the bard spell list. ("Spells Known of 1st Level and Higher")
Thus at creation you look at the bard spell list and pick two cantrips and four L1 spells off the bard list.
As you level up you can learn more and "swap out" L1+ spells you've previously picked, detailed the paragraphs following the quotes.