This is a great question, because I think the answer in this day and age is quite a bit different than it would have been even just a few short years ago.
First of all, though, if I'm going to give an honest answer here, the thing I would have to do frequently is to pray -- pray for rescue, pray for a peaceful heart and a good attitude, pray for wisdom and good decisions.
The hardest part perhaps of a situation like this is that you don't know if you'll be there for a day or two, or for the rest of your life. Knowing the answer would certainly help you establish priorities.
I suppose, though, that the things that need to be done include:
1. Before nightfall, find a place that will be safe for sleeping. A cave could work.
2. Find something to drink (fresh water)
3. Find something to eat. I'm not sure how many days it would be before I was willing to eat bugs and worms, but at least a few days ...
4. Get some sort of assessment on the geography and size of the island; climb to a high point if possible to get a good look around. This might help with establishing where you want to set up camp for the long term.
5. With all the satellites and things like Google earth, it seems that making a large sign like "HELP" or some sort of indicator that the island is inhabited would be extremely helpful. Be sure to get the sign right though. Remember the Far Side cartoon where the plane is flying over the beach, and a man on the beach is waving his arms wildly, excited at the thought of being rescued; but his sign that he drew on the beach has partially washed away, and the pilot says, "Oh never mind, it just says HELF," as the plane flies away ...
After doing that step, then at that point you may as well assume you'll be there for the long term. I'm reminded of Swiss Family Robinson, which had one fortuitous event after another. The father was remarkably adept at being able to say, "Ah, here we have the so-and-so plant; we can boil its leaves for tea and then use the fibers to make our own clothing; we can use the roots for medicine and we can use the stems to make hand woven cots."
Or something like that. :-)