What exactly should I play to best experience the writing/story?
Your initial weapon choices have zero impact on the story, they are just the tools you use to kill monsters. Different factions have different vignettes here and there but mostly the main story is the same for all characters. Pick what looks fun and don't worry about it.
I presume there is some sort of campaign questline? Is that content playable in party (and how big should the party be? (wanna grab a couple of friends to play with me)).
There are tons of missions, broken into a few different types: The Story Mission, Action Missions, Investigations, and Side Missions. A "breadcrumb" layout means that when you start out doing the first Story Mission, your path will lead you to the other types naturally. You can have several missions active at once but the Story Mission is always there. You can't rush
only the Story Mission because there are level requirements (sometimes your "mission" will be "reach level 20 by doing other missions").
Due to this structure most of the game is playable with a team. Sometimes there will be a Story Mission section where you are alone and your party can't join you (e.g. "I got kidnapped and locked in an ancient library, now I have to escape and find my friends"). When this happens each player can play through that part by themselves, and there isn't a lot of this, but it is there.
I think the party size can go up to 5? Not sure? I find a team of 3 friends is most fun. I recommend you get two friends, decide who wants to be tank, healer, and DPS guy, but don't worry too much about character builds unless you want to, just try to pick powers suited for your role and you'll be fine, the game is easy with a team. Try to rush the main Story Mission as much as possible together. Then if anyone wants to play solo they can do the other content by themselves. The Story Mission will take you like 80 hours by itself so don't worry about doing every little quest together. Also do the Investigations together and
don't look up spoilers. Investigations are the best part of the game, and solving problems brainstorming with friends is the most fun you can have here.