I have learned that all three words "embrulhar", "empacotar", and "agasalhar" correspond to the english word "to pack". However, I would like to know which one is the most common to use and whether there are differences in the meaning.
Empacotar = to pack
It can be used for packing something in a box, in a bag, etc. It's used for things that require a certain degree of protection (using cardboard, for example).
"Vou empacotar o produto e enviar para o cliente."
Embrulhar = to wrap
For things that don't need too much protection (using paper, for example). It can be used in the context of wrapping a present.
"Vou embrulhar os presentes de Natal".
Agasalhar = to put warm clothes on
Your translation of this last one is wrong, agasalhar is not to pack. Google Translate is currently showing this translation, but that is wrong.