For the Mystery Island, I found these ways so far.
Shake hands with a Sim.
Be Nice to random Sims at different times (this takes time).
Use a Workbench at different locations (if possible, and this takes time and may depend on the gained items).
Do different activities at the Park (this takes up to 5 days, unless you have a lot of Sims)
Use Google Play card money, since $50 could potentially give you enough LP to upgrade a monument to at least Level 3 by the end of the day (I tried it and it would have gone to Level 5 or higher if I were paying attention to the Houseboat requirements). This is the best method I know for if you have the money.
Other ways include just driving cars until you have enough LP to bypass the resources (pets take 1 hour minimum to find just one LP, if I remember correctly), baking cakes (still in question), having Pre-Teens interact with age-specific objects, upgrading to a Teen and having the Teen interact with things, and anything you'd normally do.
Just experiment and see what works best at which times. Sometimes you'll need to repeat activities to get it to work, so patience is necessary for once you complete the first quest.
For the Goal Resources, the quest requires you to garden, bake, read, hibernate, and something else (I forget if there's more, but I discovered a resource by accident). It can be done within 2 days if you have 6 hours a day to do stuff and at least 5 Sims available to perform actions.
Hope this helps.