You would keep your sorcerer and bard spells separate from one another. Your total spells per slot are at a 6th level caster, but your known spells are seperate. To help explain things a bit better, let me explain more in-depth.
As a sorcerer, you know 4 cantrips and two first level spells.
As a bard you know 3 cantrips and 8 spells that can be third, second, or first level spells.
Since you have multiclassed two spellcasting classes, you can cast 4 first level spell slots, 3-second level spell slots, and three third level spell slots.
You can choose these spell slots to be from either class and can even amp up some spells. For example, magic missile can be cast as a third level spell even though you only have magic missile in your sorcerer class.
TLDR: He can only take new bard spells from the bard spell list.