During the game most of your ship classes will get one or more upgrades. The next upgrade for a ship is displayed in the bottom left corner of the ship's information screen, together with a toggle to allow or disallow upgrades.
The "upgrade" text will lead you to a screen displaying the details of each of the upgrades. It shows the damage incurred, how long the upgrade will take and the minimum repair shipyard size (=/= port size). Ships also need a sufficiently large and well supported base (repair ships, naval support) for their upgrades, the higher the tonnage the bigger the port and the more support is required. This screen also shows the stats of the ship after the upgrade. If a ship has not been upgraded yet when the next upgrade becomes available, the screen for the old upgrade will show that this upgrade has been missed. The new upgrade will change its delay times, damage and port requirements to reflect that both upgrades will be done at the same time. Usually the time needed and damage done to do 2 upgrades at the same time is less than if both upgrades had been done separately.
There are ways to manage ship upgrades more quickly than just doing it one ship at a time. When looking at a list of ships, you will always have the option to turn upgrades on or off for all the ships on the list. You can also choose to only look at ships that are due for an upgrade.
Once a ship has started its upgrade, it will be put in pierside or shipyard mode. You can choose to change the mode, but the ship will not be available for task forces until the minimum delay for the repair has been finished.
A ship conversion means the class of the ship is changed. This usually involves changing ship type.
Ship conversions work roughly like upgrades, but the player always has to start the conversion manually. Possible conversions are shown in the ship information screen, right by the upgrade option. These are either in yellow or grey. Yellow means the conversion can be done right away. Being at sea, in too small or ill-equipped a port, the conversion not yet being available, etc. will all result in the conversion option being grayed out.
Clicking on one of the upgrades will show the details of the ship after the conversion, along with upgrade damage, time needed, etc. When looking at the details, the player can choose to convert to this type in the bottom left of the screen.
There is no easy in-game way to tell which ships can be converted apart from going through the ship information screens. The best tool to use for this is WITP Tracker