Learn Italian: Regalerei un viaggio alla mia amica. - I would gift a trip to my friend.

It comes from the regular -are verb regalare. To form the present conditional of -are verbs, you change the characteristic -a- to an -e- (making the stem regaler-), and then add the "I" ending -ei. So, regalerei means "I would gift."

Is regalare different from the verb dare (to give)?

Yes! Dare simply means "to give" in a general sense. Regalare specifically means "to give as a gift." Since the sentence is about treating a friend to a trip, regalare is the natural choice here.

Why does it say alla mia amica instead of just a mia amica?

In Italian, you almost always need the definite article (like la) before a possessive word (like mia). The preposition a (to) combines with la (the) to make alla. The main exception is for singular family members (like a mia madre), but since a friend (amica) isn't a family member, it follows the normal rule and keeps the article.

Do I need to include the pronoun io (I) before regalerei?