How do I say “Can I have ...” in Modern Standard Arabic?
Usually I will just say أعطيني with a questioning intonation but I am not sure if that’s right or even if that’s a little rude. How do I ask for an item politely?

Question

Answer

If you want to ask for something politely by using the ver you mentioned, you have to say:

هل يمكن ان تعطيني الكتاب؟

أعطني الكتاب من فضلك.

Answer

أعطيني it is Egyption 
If you want fusha is أعطني 
أعطني 
Not
 أعطيني

But if you say 
 أعطني من فضلك

It will be better