"I like your dress."
Translation:Podoba mi się twoja sukienka.
I understand why you thought that, the thing is that there are two options:
Either the literal translation "Lubię twoją sukienkę", which sounds rather strange without a specific context - it sounds as "of all your dresses, I have always liked this one and I have nice memories connected with it, etc". And this sentence would take biernik indeed, as "lubić" needs biernik.
Or what makes more sense here: "Podoba mi się twoja sukienka", ergo: "I like your dress" as in "I think your dress is pretty". "Your dress is pleasing to me" seems like the closest equivalent in terms of grammar. And in such a sentence, the thing that you like is actually the subject of the sentence. So it's put in Nominative.
I added "strój" in singular, but can it really be used for plural, won't in need to be "dresses" for "wasze stroje"?
In your second sentence, the verb needs to be "Podobają mi się", as you used plural 'stroje'. Podoba mi się wasz strój / Podobają mi się wasze stroje.
(owning one dress by more than one person is less probable but not impossible)
"lubić" is more general, a long-term 'liking' of something.
"podobać się" is often about the first impression, mostly the aesthetic one.
Let's change the sentence to "I like my dress" for a moment: "Lubię moją sukienkę" may be saying that I think it's comfortable, even if it's not that pretty.
"Podoba mi się moja sukienka" means that I think it's pretty. Maybe it doesn't have to be this exclusively, but most likely.
"Podoba mi się twoja sukienka" (with 'your dress') is quite likely said when you see it for the first time.
Oh, the best way to see the difference is if you make it about a dress you see in a shop: you will not say "Lubię tę sukienkę" about a dress you just saw for the first time.
Now, about "mi", it's definitely needed. It's a Dative form of "ja", so literally this sentence is something in the direction of "This dress is pleasing to me". You need to have the information 'to whom' this dress is pleasing.