This one is an easy achievement. I recommend doing this after you have made a sailing raft as it travels faster and you don't want to risk running out of food or water while doing this.
Gather at least 4 sticks on your starter island and save at your shelter. You can reload to this save if you get into trouble.
Build your raft and start sailing to the nearest island, once there plop a campfire down.
Leave and head to a new island. If you have 4 sticks build another campfire or collect the sticks on the new island. Plop down a campfire.
Rinse and repeat until the achievement pops. I was able to do this in more than 1 sitting but i have read where the achievement only popped if done in one sitting on steam.