The Magic Initiate feat allows you to pick a 1st level spell, which you can cast once per long rest. From the PHB (p. 168):
Choose a class: bard, cleric, sorcerer, warlock, or wizard. You learn two cantrips of your choice from that class’s list.
In addition, choose one 1st-level spell from that same list. You learn that spell and can cast it at its lowest level. Once you cast it, you must finish a long rest before you can cast it again using this feat.
Your spellcasting ability for these spells depends on the class you chose: Charisma for bard, sorcerer, or warlock: Wisdom for cleric or druid; or Intelligence for wizard.
From the Sage Advice Compendium’s Magic Initiate section:
If you’re a spellcaster, can you pick your own class when you gain the Magic Initiate feat? Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell.
If you have spell slots, can you use them to cast the 1st level spell you learn with the Magic Initiate feat? Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.
So, if my character is a spellcaster, we know that my character can learn a spell on their spell list, and they can use spell slots to cast it. For classes that have a number of spells known, like Bards, Sorcerers and Warlocks, this is straight forward. But what of classes that have to prepare their spells, such as Clerics, Druids and Wizards, to be able to cast them?
My question is, if I had a Cleric, Druid or Wizard, who took the Magic Initiate feat to learn a spell on their list (which they could then cast with a spell slot), would they still need to prepare the spell in order to cast it with a spell slot, or would this effectively give them an extra “prepared spell”?
If they simply know that spell because of the feat, and don’t need it prepared to be able to cast it once per long rest (via the feat), does that in some way bypass the need to prepare the spell to cast it via spell slots? This isn’t clear to me after having looked at the spellcasting class feature of such classes.