Learn Finnish: Täällä on meidän uusi linnoitus.

Both mean "here," but täällä refers to a larger, more general area or space. The word tässä pinpoints a specific, exact spot right at the speaker's fingertips. Since a fortress occupies a large area, täällä is the natural choice.

How does täällä fit into the three-way movement system we are learning?

The word täällä is the static "at" form used when something is resting in a general area. If you were moving or building the fortress to this area, you would use the destination form tänne. Because the fortress is just sitting there statically in our sentence, we use täällä.

Where is the possessive suffix on linnoitus? Shouldn't "our fortress" be meidän linnoituksemme?

In strict standard Finnish, yes, it would be meidän uusi linnoituksemme. However, in everyday spoken and informal Finnish, it is extremely common to drop the possessive suffix entirely as long as the genitive pronoun meidän is clearly stated beforehand.

Why does the sentence start with the location täällä instead of the subject meidän uusi linnoitus?

Starting with the location is the typical way to form an existential sentence in Finnish. It sets the scene first and then reveals what is there. It sounds like a proud presentation. If you started with the fortress, it would sound like you were answering a specific question about where it had gone.