My DM does not believe that you can use a Warlock's spell slots to cast other casting classes' spells at your Warlock spell slot level, except via the feat Magic Initiate. How can I convince them that you can? Where is this explained in the core books?
For example: you are a Bard 1/Warlock 5. Can you cast the Bard Cure Wounds spell using one of your two 3rd-level Warlock spell slots? Does the Cure Wounds heal 3d8 + Cha mod instead of 1d8 + Cha mod? If so, where in the PHB or DMG can I find this?
Yes, you can use warlock slots to cast any spell you know. It's in the PHB in the section on multiclassing. Page 164:
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.
The effect is determined by the slot's level, as always in D&D 5e.