Learn Italian: Do un abbraccio a mia madre.

Normally, possessives in Italian take a definite article, like la mia casa. However, there is a special rule for family members: singular family members do not take the article. That is why it is just mia madre. When we add the preposition a (to), it stays as two separate words (a mia), rather than combining into alla.

In English we can say 'I give my mother a hug'. Why is the order different in Italian?

In English, we can put the person receiving the action right after the verb and drop the word 'to'. Italian does not allow this shortcut. You usually state the direct object first (un abbraccio), followed by the person receiving it along with the preposition a (a mia madre). It literally translates as 'I give a hug to my mother'.

Why use un instead of uno for abbraccio since it starts with a vowel?

For masculine words, un is the standard word for 'a' or 'an' and is used before both consonants and vowels. You only use uno before masculine words starting with a 'z' or an 's' followed by another consonant, like uno zaino or uno stivale.

Where is the word for 'I' in this sentence?

It is built into the verb! Do is the 'I' form of the irregular verb dare (to give). Because the verb form clearly shows who is doing the action, the subject pronoun io is usually left out unless you want to add special emphasis.