Learn Spanish (Latin America): El profesor dice que habitaremos la nueva colonia. - The teacher says that we will inhabit the new colony.

It comes from the regular verb habitar (to inhabit). To form the future tense for regular verbs, you do not chop off the ending like you do in the present or past tense. Instead, you keep the entire infinitive (habitar) and add the future ending for 'we', which is -emos. That gives us habitaremos (we will inhabit).

Could I say vamos a habitar instead of habitaremos?

Yes, you could! Vamos a habitar means 'we are going to inhabit' and is very common in everyday speech. However, the future tense (habitaremos) is often used for events in the distant future or for formal predictions, which fits perfectly here if the teacher is talking about a future colony.

Why does nueva come before colonia here? Don't adjectives usually come after the noun?

You are right that most descriptive adjectives go after the noun. However, nuevo and nueva can go before the noun when they mean 'newly acquired' or 'new to us.' Placing it before colonia emphasizes the idea of this being their next big venture, rather than just describing the physical age of the colony.

In English, we can say 'The teacher says we will...' without the word 'that'. Can we drop que in Spanish?

No, you cannot drop it in Spanish. Whenever you connect two phrases like this (the teacher says + we will do something), you absolutely must use que. Saying El profesor dice habitaremos is grammatically incorrect.