"Give me the red trainers please."
Translation:Дайте, будь ласка, червоні кросівки.
I thought дайте meant give me. So wouldn't дайте мені mean give me me?
No. Strictly speaking, дайте is the verb "to give" in vocative form. Technically, it should be дайте менi "give [to] me," but the менi is often left off in informal conversation, as it's implied that the person who is asking wants you to give the object to them, unless they specify a different recipient.
If you really wanted to say the unlikely sentence of "give me me," as in 'give me myself,' it would be дайте менi мене, or "give to me me," where дайте is the vocative form "give!", менi is the dative form and indirect object "to me," and мене is the accusative form, "me," and also the direct object in this sentence. Hope this helps.