You may have noticed that there are two types of expeditions - rp and random. Rp expeditions have a limited number of participants and slots are first come first serve. You sign up with a sam to get a slot and then rp your sam going on the expidition. Whomever is running the expidition will let you know if your sam has discovered any items as they explore. Everyone gets at least one item for participating (I think) but there are usually more items give out and occasionally some other sort of surprise.
The random expedition is open o everyone and doesn't require that you sign up in advance. Its generally a pick a door styles with everyone allowed to claim on slot every few hours or so until all the slots are gone. Last time there was a cap on the total slots someone could claim. There are a few items hidden behind the doors and, as with the rp expedition, occasionally some other sort of surprise.
I've never (yet) managed to get a slot in an rp expedition but I've done two of the random ones. They are really fun and I liked the last one cause it had a little gridded picture that you could 'search' in I haven't gotten anything yet but hopefully I'll be online when slots open for the next rp expedition.