As an example, A level 6 bard multi classes to a level 1 Cleric; does the bard have the spell slots that come with level 7 (Being 1 4th level spell slot) or would the bard have to wait until they level into level 7 before getting that spell slot?
When you multiclass you are a level 6 Bard and a level 1 Cleric.
You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.
6 bard + 1 cleric = 7 spellcaster levels
7th Level: 4 1st, 3 2nd, 3 3rd, and 1 fourth.
Now, you might not know any fourth level spells, but even if you don't you can spend that slot to cast lower level spells at the fourth level -- possibly taking advantage of a spell that has greater effect when cast "at higher level".