Best way to play co-op?

I am a longtime pokémon fan, but I intend to only play with my GF as we enjoy co-op games and hope to replicate the wonderful experience we had on Stardew Valley.

We'll play through Game Share owning one NS2 and one NS.

I have read that the "story mode" can't be played in co-op, and that the "guest" player will be "restricted" when visiting the host.

I'd be grateful to anyone helping me understand the extents of these restrictions, and whether we should play using the "online island" feature or just normally as a guest.
Any help related to my concerns will be so appreciated, especially if you personally experienced co-op as we wish to do.

Thank you!