I have a Bard 4/Sorcerer 1 that is going to continue levelling in the bard class. When he reaches Bard 5/Sorcerer 1 and he learns his first 3rd level spell, can he take from the sorcerer list or just the bard list?
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.
You choose from the bard list
The rules for spellcasting when multiclassing are here.
As it says:
Spells Known and Prepared. You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
Generally, when levelling up you just follow the standard rules for the class you are levelling up in. The multiclassing rules indicate the few exceptions (such as how many spell slots you get).