Learn French: La nouvelle récolte est vraiment meilleure.

Because meilleur (better) is an adjective, it has to match the gender and number of the noun it describes. Since la récolte (the harvest) is feminine, we use the feminine form meilleure.

Could I say "plus bonne" instead of meilleure to mean "better"?

No, you can never say plus bon or plus bonne in French. Just like in English we say "better" instead of "more good", French uses the special irregular adjective meilleur (or meilleure for feminine words).

When do I use mieux for "better" instead of meilleur?

Use meilleur (and its feminine/plural forms) when you are describing a noun, because it means "more good". Use mieux when you are describing a verb (an action), because it means "more well". Since we are describing the noun récolte here, we use meilleure.

Why does nouvelle come before récolte instead of after it?

Most French adjectives go after the noun, but nouvelle (new) is an exception. It belongs to a small group of common adjectives related to Beauty, Age, Goodness, and Size (often remembered by the acronym BAGS) that always go before the noun.