The first requirement is that you have a colonization ship built or the free one you get when you discover your first island. You need to activate it from the inventory bag at the lower right.
Now click on Island Overview (Palm Trees, lower right) and then click on the Ship icon in the upper right of that screen.
Scroll down and locate your colonization ship from that list which looks like a house and will have the status of idle. Click on that ship.
Now click on the island that you want to colonize. Ensure that you have an island slot marked Discover in your 3x3 Island listing.
You then click on the colonize button at the bottom of the Islands listing. The timer under that island will change to 5 hours and count down from there.
When the colonizing ship arrives at the island it will appear in your Islands 3x3 matrix and now be available for development. Make at least one lumberjack hut right away.
The colonizing ship is consumed in the process of that act so you will need to build one for each future colonizing event. You will get a harbourmaster complete with the “dock” aka transport storehouse. Make sure you have one on Newport as well or you cannot do transports.
Build a warehouse as soon as possible and upgrade it as soon as those become available and you have the appropriate resources. A Distribution House is optional for a new island since the harbourmaster is one as well. Click on the harbourmaster to view that coverage range. As you open sectors, they will be added. If there is any doubt as to whether you have proper coverage, double click the harbourmaster to see your island distribution network. Any building that is red has a problem and needs to be corrected. Often the problem is a bad road connection or DH out of range.