The Italian verb for “to give” is dare. But if you want to give someone a gift, you need to get used to using the verb fare (to make)—fare un regalo:  

Quella gli faceva un regalino, quell'altra l'invitava a cena...

One would give him a little gift, another would invite him to dinner...

Caption 49, Il Commissario Manara - S1EP3 - Rapsodia in Blu

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Or you can use the verb form of regalo, regalare:

Eh, ma mi sa che questo è l'ultimo anno

Uh, I guess this is the last year

che ti posso regalare le mie scarpe.

that I can give you my shoes.

Caption 4, Un medico in famiglia Stagione 1 - EP1 - Casa nuova

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Regalo is analogous with “present,” and it’s the word you will be using most of the time. However, another way to say “gift,” which often implies a divine or important giver, is dono. You’ll hear it in conjunction with traditions, and indeed, dono is used like regalo in talking about what Santa Claus brings down the chimney.

Ovviamente ai bambini portava doni.

Obviously to children he brought gifts.

Caption 16, Marika spiega - La vera storia di Babbo Natale

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Donare is easy to remember, being very similar to “donate.” In fact, as a verb, donare can mean “to donate,” as in money or blood: donare sangue (to give blood). Blood donors are donatori di sangue.

Of course, gifts are not always tangible.

Perché io lavoro con un grande dono prezioso che ognuno di noi ha...

Because I work with a precious gift that each one of us has...

Io lavoro con la mia voce.

I work with my voice.

Captions 7-9, Marika e Daniela - Daniela Bruni, voice over

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And now you need to stretch your mind a bit because the giver is an item of clothing. The shirt in question gives the wearer some positive quality. This particular use of donare is worth remembering because it’s a wonderful way to compliment someone! (Note that the person is using the polite form; to a friend you would say ti dona.)

Ah... ma lo sa che questa camicia le dona?

Ah... you know that this shirt looks good on you?

-Grazie. -Sì. Fa esaltare il colore dei suoi occhi.

-Thanks. -Yes. It brings out the color of your eyes.

Captions 8-9, Il Commissario Manara - S1EP2 - Vendemmia tardiva

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Finally we have dote (talent, gift, quality) with its verb form, dotare. Except for when it means “endowment” or “dowry,” dote isn’t the kind of gift you can give someone. 

Il ragazzo è dotato per la musica e sua sorella invece è dotata per il disegno.

The boy is a gifted musician while his sister is a gifted artist.

Ha una dote per la musica.

He has a gift for music.  

We could say that God, or some higher being has “provided” that boy with his gift for music. So don’t be surprised if you go to buy a TV in Italy and the salesman tells you that la TV è dotata di telecommando (the TV is supplied with remote control). Not God-given, but factory-given!

To sum up on a practical level (leaving Christmas, weddings, and TVs aside):

• Il Regalo and regalare have to do with worldly things for the most part, things one person can give another. Regalo can be synonymous with “present.” Remember to use the verb fare with regalo! 
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• La dote and dotare have more to do with talent, and you can’t give someone this kind of gift.
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• Il dono and donare mostly give the idea of a bigger picture: a gift from God, a donation to a cause, donating blood or organs. But donare can also mean “to enhance,” to bring out positive qualities.

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Learning suggestion:

What are your natural talents or gifts? What about those of your family and friends? What did you get for a present on your last birthday? Do you know people who give blood? What are the earth’s natural gifts? Make a list of what comes to mind and then choose the Italian word that is closest in meaning.

To test out any phrases you come up with, just Google them and you will probably get some clues. If you have doubts, use WordReference or other dictionaries to get some more complete input than this lesson can provide.