из = from the inside out.
"Take the eggs from the basket, please" = "Возьми яйца из корзины, пожалуйста".
"I am from Russia" = "Я из России".
от = the direction (nothing was inside).
"Get away from me" = "Отойди от меня".
"The letter goes from Anna to Dan" = "Письмо идёт от Анны к Дэну".
If you are totally sure that the subject cannot be inside something (like "gift from me", well, the gift may be inside your body only if you are pregnant woman! :D), use "от".
о/об/обо, на, при, and в. I agree that it was a bit puzzling at the beginning to find that it is called "prepositional", but restricted to those. Then again, I think it is the only case that is used only together with prepositions, where words in other cases can appear without one.
So, I frequently mistype the English word "to" as "ot." I wonder if Russians ever frequently mistype the word "от" as "то." If so, I find amusing that I type my "to" as their "from," and they type their "from" as my "to."
Potentially just for my amusement only. But it will help me remember the meaning of от.