Eberron: Rising from the Last War has rules for multiclassing as an artificer on page 54:
Optional Rule: Multiclassing
If your group uses the optional rule on multiclassing in the Player’s Handbook, here’s what you need to know if you choose artificer as one of your classes.
Ability Score Minimum. As a multiclass character, you must have at least an Intelligence score of 13 to take a level in this class, or to take a level in another class if you are already an artificer.
Proficiencies Gained. If artificer isn’t your initial class, here are the proficiencies you gain when you take your first level as an artificer: light armor, medium armor, shields, thieves’ tools, tinker’s tools.
Spell Slots. Add half your levels (rounded up) in the artificer class to the appropriate levels from other classes to determine your available spell slots.
(Eberron: Rising from the Last War pg. 54)
(For the record, these are the same rules that the last Unearthed Arcana artificer had prior to the publication of Eberron: Rising from the Last War.)
As you see here, spell slots are handled by halving your artificer levels, rounded up, unlike most half-casters. Otherwise, it works just the same as for other spellcasters: after you halve your levels (rounded up), and add that your levels (scaled appropriately) from other classes, find that number on the multiclass spellcaster table and that’s your spells per day.
Multiclass spellcasters don’t do anything special with cantrips: you have your cantrips from the cleric class levels, and your cantrips from your artificer class levels, and your levels in the other class doesn’t affect either. So you never get more cleric cantrips by taking artificer levels, and you never get more artificer cantrips by taking cleric levels.
The artificer’s ability to swap cantrips applies only to artificer cantrips (“one of the artificer cantrips you know”), and only when you take a level of artificer (“When you gain a level in this class”) (Eberron: Rising from the Last War pg. 55). It never lets you modify your cleric cantrips, and you don’t get to do it when you take a cleric level, only when you take an artificer level.
(The last Unearthed Arcana artificer instead had the ability to change one cantrip after each short or long rest—that ability is no longer found in the Eberron: Rising from the Last War version of the artificer, but even in the Unearthed Arcana version it was still limited to only modifying artificer cantrips.)