There are (at least) two words which mean "to give": あげる and くれる. The former is used when you give to someone else, and the latter is used when someone else gives to you.
But which do you use if you want to say that someone else gave something to another person, where both parties are outside your social circle?
That would definitely be 「あげる」 if you are speaking from the giver's standpoint.
「AさんはBさんにプレゼントをあげた。」= "A gave B a present."
If, however, you are speaking from the receiver's standpoint, you would use 「もらう」.
「BさんはAさんに（or から）プレゼントをもらった。」 = "B received a present from A."
「くれる」 cannot be used to talk about a transaction between two third parties.
For your situation you will want to use あげる, which is used when the recipient is not in your social circle. The giver can be anyone.
Take note of social status too. If the recipient is socially superior the giver, then you should use 差し上げる（さしあげる), but you don't have to. When the recipient is socially equal, or slightly lower, then you use あげる。 If the recipient is inferior you can choose to use やる, but that is often only for pets, plants, or your in-group inferiors (such as younger siblings).
田中さんは山田先生に本を差し上げました／あげました。 Mr. Tanaka gave Mr. Yamada (social superior) a book.
山田先生は田中さんに本を上げました。Mr. Yamada gave Mr. Tanaka (social inferior) a book.
田中さんは鈴木さんに本を上げました。Mr. Tanaka gave Mr. Suzuki (social equal) a book.
田中さんは子猫にえさをやりました。Mr. Tanaka gave the kitten (social inferior) some food.
Note: くれる can be used when the recipient is in your social circle (your family), not just yourself. If you (or someone in your family) receives something from a social superior, you have to use くださる.
山田先生は母に本をくださいました。Mr. Yamada (social superior) gave my mom a book
You need to refine your thinking on these verbs. Rather than thinking of あげる as you give to someone else, think of あげる as anyone giving something to anyone else except for when the recipient is you/your in-group.
Then you adjust for register (さしあげる vs あげる, etc.), but within a single register, that's how you cover all your "to give" bases with あげる／くれる。
