If I start with a Celestial Warlock with Cure Wounds and then multiclass into Bard, can I cast Cure Wounds as a Bard now, or do I have to learn the spell a second time to be able to cast it as a Bard?
If only one class knows/can prepare the spell, it is only associated with that class
The section in the PHB about spellcasting while multiclassing will help with much of your confusion.
When you multiclass as a spellcaster, the rules change slightly with regards to how your spells are prepared/known and cast.
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell.
Let us assume that you only know cure wounds from your class in warlock and that you never learned it from bard. In this case, when you cast cure wounds it is always considered to be cast as a warlock spell. What this means is that it will use your warlock's spellcasting modifier and spell save DC and any other qualities that affect the spell you are casting. Also it would qualify for any features that relied on something being a warlock spell but not be able to take advantage of anything requiring it be a bard spell.
However, you share slots and Pact Magic slots between the classes
One place where the spells do overlap is in what slots you can use to cast them.
If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.
Thus you can cast cure wounds (still as a warlock) but using any slots that you have available. For example, as a Warlock 1/ Bard 3 you would have 5 first level slots and 2 second level slots available to cast any spell that you know. Which means you could cast that cure wounds using a second level slot if you wanted.