You can use «мне», however, you can omit the first person pronoun here, too. Especially if asking for an item at the store. Of course, it is a different story if the item is supposed to be given to someone other that you :).
Alternatively, you may omit "дайте" in colloquial speech or say "можно" (possible) instead:
- Мне курицу, пожалуйста.
- Мне кофе, 2 чая и чизкейк.
- Можно 2 латте?
- Можно мне курицу и рис?
That is not to say that we accept such structures as a tranlsation of "gve me" (which, in English, is not a good way to place an order anyway). Personally, I do not use «можно» much in these situations but I use "мне"-phrasing rather often.
Тарелка is the Nominative form. When an object is a direct object of a verb (memorise it on per-verb basis, actually) it takes a different form. Long story short, nouns ending in а/я (e.g., мама, кошка, тарелка, Россия, змея, семья) have a unique form for that purpose in the singular:
- Я знаю маму
- Я знаю эту кошку.
- Я знаю Россию.
Words like стол (masc.) or яблоко (neut.) re-use the dictionary form if they are inanimate—or re-use the Genitive if they are animate. In plural, this extends to all nouns whatsoever.
It is дать. The congugation is somewhat similar to the conjugation of есть:
- Я дам. = I will give
- Ты дашь. = You will give (sg.)
- Он даст. = He will give.
- Мы дадим , Вы дадите, Они дадут. (for есть it is "они едят", otherwise the forms are similar)
This verb is perfective, so its conjugated non-past forms express the future action.
Note that the imperfective давать is more regular. It belongs to a small unproductive class of verbs that includes those ending is -давать, -ставать and -знавать. There is nothing special about this class except that its present tense stem loses -ва and inserts the й sound instead: я даю, встаю, узнаю / ты даёшь, встаёшь, узнаёшь etc.
I tried "Pass the plate, please" (which was some nice alliteration), but it didn't like it. I know in the strictest definition, "pass" and "give" aren't quite the same - but in any reasonable context, "pass the plate" and "give me the plate" are functionally identical, and - at least in the UK - I would argue that "pass" is a lot more common, and sounds less rude and imperative.