Learn Icelandic: Í fyrsta lagi þurfum við að tala saman.

Lagi is the dative form of the neuter noun lag, which has many meanings including "layer", "shape", or "tune". In this fixed expression, it functions like "place" in the English phrase "in the first place". You will also see this pattern continue with í öðru lagi (secondly) and í þriðja lagi (thirdly).

Why is the verb þurfum placed before the subject við?

This is because of the V2 (verb-second) rule. In Icelandic, the main verb must usually be the second grammatical element in a main clause. Since the prepositional phrase Í fyrsta lagi takes up the first slot, the verb þurfum must come immediately after it, pushing the subject við into the third slot.

Why is the adjective fyrsta used here instead of the base form fyrstur?

The preposition í takes the dative case here, and lag is a neuter noun. Fyrsta is the weak, neuter, dative form of fyrstur (first). In fixed expressions pointing to a specific, definite order (like "the first place"), Icelandic requires the weak adjective form.

Could I just say þurfum við að tala without adding saman at the end?

You could, but adding saman (together) is the most natural way to express having a mutual conversation, much like "talk with each other" or "talk things out" in English. Without saman, it can sound a bit more abrupt or one-sided, like a boss or parent saying "we need to speak".