"I don't like this wine."
Translation:No me gusta este vino.
When a verb requires an object pronoun (direct or indirect), any negatives (no, nunca, etc) must precede the object pronoun. See https://www.spanishdict.com/guide/direct-and-indirect-object-pronouns-in-spanish
"Mi" literally means "my", so it would only be used when you're expressing possession of something. For example, "Es mi mochila" -> It is MY backpack. Or, "Ella es mi amiga" -> She is MY friend.
On the other hand, "me" is reflexive. It can be used with the verb gustar to indicate something is pleasing to you. So, if you say "Me gusta vino" it's really like saying "Wine is pleasing to ME" in English. "Me" can also be used in reflexive verbs where you express doing something to yourself. For example, You can say "Voy a ducharme" --> I'm going to shower MYSELF.
I hope this makes sense!
The link from Eric about pronouns is beyond me. I don't understand how that helps to explain why I cannot say "Yo no me gusta este vino". Why is it wrong to include the "Yo"? Is it supposed to be "No yo me gusta..."? That doesn't seem right. Can someone explain that in basic terminology?