Most of the things you need to do on the Welcome Island are covered in the many excellent general guides, so read either
this Citizenship walkthrough page or
this general getting started page first. In this page however, only specific tips are listed, assuming you have already read and understand the basics.
Legacy
When your character gets near the point of running out of time, you need to begin considering what you will do with him/her when they pass on. All the buildings have an ownership attribute that can be transferred, so you are able to pass along all of your hard work to another, future character. Most importantly, you need to create a new character and get them to the mainland before the current character's time runs out. Having done that, you can transfer all the buildings and save your next character that much work, thus skipping forward to future progress.