Learn Polish: To będzie spokojna emerytura. - This will be a peaceful retirement.

You are exactly right that the verb być (to be) usually takes the Instrumental case when saying "X is Y" or "X will be Y" (for example, On będzie dobrym dziadkiem - He will be a good grandfather). However, there is a very important exception: when the subject of the sentence is just the pronoun to (meaning "this" or "it"), the noun phrase that follows stays in the basic Nominative case.

Since emerytura is a feminine noun, shouldn't we use ta instead of to?

If you were saying "This retirement will be peaceful", you would indeed use ta (Ta emerytura będzie spokojna) because "this" is acting as an adjective attached directly to a feminine noun. But in our sentence, to acts as a standalone subject pronoun meaning "this" or "it" (as in, "This will be..."). When used this way, it is always the neutral to, regardless of the gender of the noun that comes later.

Does the meaning change if I drop the word to and just say Będzie spokojna emerytura?

Yes, it changes the meaning slightly. In Polish, dropping the subject often turns the sentence into an existential statement. Będzie spokojna emerytura translates closer to "There will be a peaceful retirement." Keeping to makes it clear you are pointing out a specific situation: "This will be a peaceful retirement."

How do I know which future tense form of to use here?