Learn Spanish (Latin America): La tarde está agradable.

We use está (from estar) because we are describing how the afternoon feels right now as a temporary condition. If we said La tarde es agradable, it would describe a permanent characteristic, meaning that this afternoon is fundamentally a pleasant one or that afternoons in general are pleasant.

Since tarde is feminine (la tarde), shouldn't the adjective end in "a"?

The adjective agradable ends in -e. In Spanish, adjectives that end in -e (like grande or inteligente) use the exact same form for both masculine and feminine nouns. You only change them by adding an -s if the noun is plural (like las tardes están agradables).

Could I drop la tarde and just say Está agradable?

Yes! If it is already clear from the context that you are talking about the weather or the afternoon, you can drop the subject entirely and just say Está agradable (It is pleasant).