Learn Russian: У дяди и тёти есть собака.

This sentence uses the Russian "to have" structure, which literally translates as "By the uncle and aunt exists a dog." The preposition у (by/at) requires the nouns that follow it to be in the Genitive case. Since дядя and тётя both end in -я, their endings change to -и in the Genitive.

Isn't дядя (uncle) a masculine word? Why does it get the same ending as the feminine word тётя (aunt)?

You are right, дядя is grammatically masculine because it refers to a male. However, because it ends in -я, it follows the exact same spelling rules for its endings as feminine nouns that end in -я. That is why both words change to дяди and тёти.

Why is собака in its basic dictionary form? Shouldn't it change because it is the object they have?

In English, "dog" is the object. But remember the literal Russian translation: "At the uncle and aunt exists a dog." In Russian, the dog is the one doing the "existing." This makes собака the subject of the sentence, so it stays in its basic Nominative form.

Do I always have to use the word есть (is/exists) when saying someone has something?