"I have always wanted to be rich."

Translation: Zawsze chciałem być bogaty.

[luckyarii]

I just need some clarification on when to use chcialem and chcialam. Since I'm a male, I use chicalem, right?

[immery]

Yes. -łam is for females- łem is for males

[NikoUlven]

Yes

[Jane308391]

Please, why is bogaty not in the instrumental?

[Jellei]

Only noun phrases take Instrumental after "być". If it's just an adjective, as here, it stays in Nominative.