Learn Hardcore Turkish: Limon taze.

In Turkish, the 3rd person copula is normally omitted in simple statements. Instead of saying “The lemon is fresh”, you just put the subject (Limon) and the predicative adjective (taze) together: Limon taze.

Why does the adjective come after the noun here? Aren’t adjectives supposed to precede nouns in Turkish?

When an adjective directly modifies a noun (attributively), it comes before the noun (e.g. taze limon = fresh lemon). In Limon taze, however, taze is a predicate adjective describing the subject. Predicative adjectives follow the noun in Turkish.

There’s no article like a or the. How do I express definiteness or indefiniteness?

Turkish has no articles. A bare noun can be indefinite, definite or generic depending on context.

Limon taze can mean A lemon is fresh or Lemon is fresh in a general sense.
To say this lemon is fresh, you add a demonstrative: Bu limon taze.
To say the lemons are fresh, you make it plural: Limonlar taze.

How would I negate Limon taze to say “The lemon is not fresh”?

You insert the negative copula değil after the adjective:
Limon taze değil
This literally reads “Lemon fresh not,” which means “The lemon is not fresh.”