Learn Russian: Я люблю моего доброго дедушку.

Дедушка (grandfather) is a masculine noun that happens to end in -а, so it changes to дедушку in the accusative case, just like a feminine noun would. However, words that describe it (like мой and добрый) still treat it as a regular masculine noun. Since the grandfather is alive (animate) and is the direct object, its modifiers take the masculine animate accusative endings -его and -ого.

How is the letter г pronounced in моего and доброго?

In the masculine and neuter adjective endings -ого and -его, the letter г is always pronounced like an English 'v'. So моего sounds like 'ma-ye-vo' and доброго sounds like 'do-bra-va'.

Why doesn't the masculine animate noun дедушка take an -а ending in the accusative, like брата (brother) does?

Regular masculine nouns ending in a consonant take -а or -я when they are animate direct objects. But masculine nouns that already end in -а or -я in their dictionary form (like дедушка, папа, or дядя) follow the feminine declension rules for the noun itself. So the -а changes to -у.