Say I play a wizard who learns cure wounds by taking magic initiate. Could I cast that spell using the Wizard's spell slots?
No. You are not a Cleric so you can't use a spell slot to cast the 1st level spell learnt from the Magic Initiate (Cleric) feat (or whatever class you picked for the feat).
From the Sage Advice Compendium:
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.
If you think about the source of where the magic is coming from for each magic user, you can see that it just doesn't make sense for you to use higher level spell slots.
For example, a wizard can't call upon a cleric's deity to grant him more power, and a bard can't call upon a sorcerer's demonic entity for more spells.
The Sage Advice Compendium appears to be misquoting it's own source texts (which is truly frustrating because the most indelible measure of being wrong is being wrong by your own logic) as, in the PHB, the Sorcerer Spellcasting feature makes no mention of specifically casting "Sorcerer" spells.
The above ruling is wrong, at least in the 5e rule set as written (including errata). After a careful reading of the spellcasting ability description of all classes you will find the specificity of the cleric description to be an exception in terms of wording.
Further examination of the multiclassing section of the PHB will net you the understanding that spell slots are a feature tied to Character Level, though the ability to cast spells in general using a spell slot requires having a class with the spellcasting feature, as they are the only classes with spellslots to use.
Looking through the Spellcasting chapter of the PHB will also reveal that regardless of the source of the magical knowledge, all spells follow the set out rules in the same way. In this section there is no mention of spell slots being class specific, and evidence would lead one to conclude that the function of spells being denoted as class specific pertains to the traditions in which they are learned and thus the ability modifiers applied to them.
The only difference in casting type is in the form of the distinction made between the Spellcasting and Pact Magic class features.
Add to this the fact that the only specification in the Magic Initiate feature regarding learning the spell is that it is learnt as a spell from a specific class which means the ability modifiers added to it are derived from that class' specification for spell modifier and it is obvious to me that RAW allows the spell to be cast with any spell slot, at least before the question was answered by the Wizards Team in which they misquote their own material.
This leads me to believe that either the team misread their personal notes on the topic and substituted them for final copy materials to reference, or that they need to issue another errata, or, finally, that given that the DM is the be all and end all of rules and that all forms of common sense from the material given would lead you to deduce that you can cast a spell with any spell slot because there is absolutely no mention of spell slots being class specific, most DMs should probably ignore the above ruling.
This is the way I have been running my games since we moved to 5e at least, and its the one that makes the most sense deriving from the base texts only. Though the team may have notes that clarify this, they have not shared them, and the SAC quotes a non-existent line of text, so I'm not entirely sure this ruling will stand for long anyway.
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.