Once you complete all tasks and get all stars, you will complete the chapter and you will move onto the next one. Unfortunately this also means that you must start over from scratch with a brand new raft.
But, to offset this, the game will often reward you with an additional survivor, which means you have more helping hands to go around on the raft. With more survivors, your raft can run more smoothly thanks to the extra help.
Your survivors will automatically take care of tasks aboard the raft over time. They will build things when you issue the order and harvest item structures when they are ready. They will NOT, however, swim off the raft to grab driftwood, so you have to do that manually.
You can queue up retrieval actions if you spot multiple items in the ocean. Tapping on an item will make it start flashing, meaning it is in the queue and someone will go fetch it shortly.