Example:
Sally makes Seza n half elf wizard(enchanter) folk hero from Shadowdale. So to start she gets Uncle Trapsnaper's Book of Spells, a bunch of common spells in my world. She also gets the smaller Elflore book, Enchantment book, folk hero and Dalelands books. Giving her a mix of about 200 spells her character knows about. The player can still only pick three cantrips and six at first level spells to put in the characters spell book.
The groups first adventure is Lost Mine of Phandelver. Seza finds a spell scroll or spellbook every so often in treasure or from defeated foes. And a couple goblin, drow and 'evil' type spells. When the character finds a physical copy of a spell, the player gets a copy of the spell write up.
Each time Sally levels up Seza, she can pick from all the spells she knows for the spells to put in Seza spell book.
At some point the group will choose to have down time. Seza can spend downtime researching any common spell she has seen or heard of in game. Or she can do research on common spells of a 'type', such as race, background or location. This is only for the knowledge of the spell, the player gets the spell write up. To add the spell in the game the character must still make a scroll, add it to her spellbook or such.
Uncommon spells are a bit harder. Not only does this research talk longer and cost more, but it has the additional set component. The character must find a set source for the spell. The most obvious is another spellcaster with the spell, though there are other ways.
Rare and above spells are always treasure.