Would a Cleric 1 / Wizard 1 multi-classed character know 3 cleric cantrips and 3 wizard cantrips?
Yes, you would know 3 Wizard cantrips and 3 Cleric cantrips. The Cleric class says:
At level 1, you know three cantrips of your choice from the cleric spell list.
And the Wizard class says:
At 1st level, you know three cantrips of your choice from the wizard spell list.
Finally, in the multiclassing section, we have:
You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.
Cantrips are spells (see here), and are covered by this rule.