Guys thank you so much!!!

American English - all two ways are correct and understandable

(British English) In the particular example that you give, I would expect 'I have picked up...' to always be the best verbal form to us. However, Holly's various use of 'grab' in other contexts would also occur in British English (although perhaps less frequently)

I speak Canadian English. You can absolutely say "grab" as an informal alternative to "take". Sometimes, but not always, "grab" implies a quicker action than "take". I use this term all the time in sentences like this: I grabbed my coat and ran out the door. (I use "grab" because it implies that I was in a hurry) I will be there in a minute; I will just grab a drink first. (again, hurrying is implied) Making scrambled eggs is easy. Just grab some eggs and fry them until cooked. (No hurrying implied here, but "grab" is used as an informal alternative to "take")

In English we are more likely to say 'I have picked up the cup.....'. If you have moved a cup from one place to another you could say 'I have taken the cup from the (cupboard) and now I am going to ......'. We would only use grab if you have taken it forcibly from someone's hand - 'I grabbed/snatched the cup angrily from his hand ........'