Learn Russian: Бабушка подарила мне серебряную ложку.

The verb подарить specifically means "to give as a gift." If you used дала (from дать), it would just mean she physically handed the spoon to you, like asking someone to pass a utensil at the dinner table, completely losing the context of it being a present.

Why is the word for "me" (мне) in the dative case here?

Мне acts as the indirect object—the person receiving the action. In Russian, verbs of giving, showing, or telling almost always require the dative case to show to whom the action is directed.

Ложку and серебряную are both feminine words in the accusative case. Why do their endings look so different (-у vs -ую)?

Nouns and adjectives use different declension patterns. For feminine words in the accusative case (direct objects), nouns ending in -а change to -у (so ложка becomes ложку). Adjectives, on the other hand, take the ending -ую (or -юю for soft stems), which gives us серебряную.

Why does серебряную (silver) only have one н in its suffix? I thought many adjectives used -нн-.