When do I get to upgrade my ship?

To upgrade your ship, you'll need Upgrade Modules. The amount you need is 2, 4, 6, 8, then 10 respectively for each upgrade. These can be found in chests, reward bags, and rarely as monster drops.

You must then obtain a ship license. This can either be done by getting Crew Members (2, 4, 6, 8, 10) for completing quests for settlers and hiring penguin mercenaries (after defeating Dreadwing), or alternatively by buying the license from The Penguin Bay at the outpost for increasing (10,000, 20,000, 40,000, 80,000, 150,000) amounts of pixels:

Once you have the upgrade modules and the license (fake or real), you'll need to visit Penguin Pete at the outpost, who will carry out the upgrade: