Learn Spanish (Latin America): Mi primer viaje es en enero.

Primero (first) drops the final -o and becomes primer when it comes directly before a masculine singular noun. Since viaje (trip) is masculine and singular, we use primer. (The exact same rule applies to tercero becoming tercer).

Why isn't enero capitalized?

In Spanish, the months of the year (like enero, febrero, etc.) are written with lowercase letters unless they appear at the very beginning of a sentence.

Why do we use es instead of está here?

We use es (from the verb ser) because we are stating when an event takes place. Ser is always used to tell when or where an event (like a trip, a party, or a meeting) happens.

Can I put the ordinal number after the noun, like mi viaje primer?

No, ordinal numbers (like first, second, and third) almost always go before the noun in Spanish. You must say mi primer viaje.