Learn Russian: Мой папа молодой, а дедушка старый.

In this sentence, we are using adjectives (молодой - young, старый - old) to describe someone's age, just like any other physical trait. You only use the Dative case when you are stating the exact number of years, like in мне 20 лет (I am 20).

Папа and дедушка end in -а. Doesn't that make them feminine?

Usually, nouns ending in -а are feminine. However, because папа (dad) and дедушка (grandpa) refer to male people, they are strictly masculine nouns. That is why the adjectives describing them take the masculine endings -ой and -ый.

Why is the word а translated as "and" here instead of и?

The Russian word а is used to show a contrast between two things. Here, we are contrasting the dad's youth with the grandpa's old age. It can be translated as "and" or "but" in English, but it always signals that a comparison or contrast is happening.

Where is the word for "is" in this sentence?

Russian doesn't use a present tense word for "is" or "are". You just put the subject and its description side by side: папа молодой literally means "dad young".