When you get enough crew members (or buy yourself a shady license from coat) Penguin Pete should give you the quest for upgrading it. For every two crew members you recruit, you will get this opportunity to upgrade. To upgrade your ship, you will need to recruit a crew. The amount you need is 2, 4, 6, 8, then 10 respectively for each upgrade. To upgrade your ship, you'll need Upgrade Modules. (or replace the number with the tier of ship you want) then use the item to upgrade your ship.