What is the spell save DC and spellcasting modifier for a Rogue with Magic Initiate?

Question

As a rogue I'm planning to take Magic Initiate with warlock class. The Magic Initiate feat says:

[...] Your spellcasting ability for these spells depends on the class you chose: Charisma for warlock [...]

• cantrip 1: Booming blade or Green flame blade
• cantrip 2: Minor illusion
• 1st-level spell: Hex

My stats bonuses are: STR : -3 | DEX +6 | INT 0 | WIS +1 | CHA +4 | prof : 4

Questions:

• Green flame blade :

[...] The second creature takes fire damage equal to your spellcasting modifier. [...]

What is my spellcasting modifier? 0 from INT as rogue or +4 from CHA as warlock?

• Minor illusion :

[...] If a creature uses its action to examine the sound or image, the creature can determine that it is an illusion with a successful Intelligence (Investigation) check against your spell save DC. [...]

What is my spell save DC? 12 (8 + 0 from INT + 4 from prof) or 16 (8 + 4 from CHA + 4 from prof)?

Answer 1

Your spellcasting modifier for those spells you cast using Magic Initiate is Charisma

You quoted the relevant text:

Your spellcasting ability for these spells depends on the class you chose : Charisma for warlock

So your DCs, to hit modifiers, etc. for those spells are all based on your Charisma, so: DC 16.

If you later learn those same cantrips from another source (multiclassing, Arcane Trickster roguish archetype) you can choose which ability score to use, if that second source uses a different ability score for casting. If you have the Arcane Trickster archetype, then your spellcasting ability for your rogue spells is Intelligence.

Answer 2

Charisma is your spellcasting ability modifier and is used in the calculation of your spell save DC

As you rightly pointed out, picking Warlock for the Magic Initiate feat means that Charisma is your spellcasting ability modifier for the spells you picked.

So your spellcasting ability being charisma, your spellcasting ability modifier is going to be +4.

A spell's spell save DC is calculated as 8+ Spellcasting ability + proficiency bonus, so in your case it will be 16.

This applies only to the spells you obtained through the Magic Initiate feat, and not to other spells you might have obtained through other means such as the Arcane Trickster class.

Note that if you were to take the Spell Sniper and pick Cleric (for instance) you would use wisdom and not charisma as your spellcasting ability for those new spells.

There is no real limit on the number of spellcasting abilities you may have, but it is always determined by the feature which granted you the spell.

Note that it is possible to receive access to a single spell from several sources, each with their own spellcasting ability.