To visit a tavern, enter the town by docking a ship or having a building already established, then highlight the tavern (they are the same structure in every town and are usually placed by the docks) and press to open the tavern menu. (Be sure you have opened the tavern menu and not a nearby building's menu.) You aren’t required to do anything other than open the tavern menu. There are 60 towns on the map so you need to visit them all. Click HERE for a map showing all 60 towns.
TIP: You may visit all taverns as you progress through the game but it is easy to lose track of where you have been and it will take some time to sail around the Caribbean. A good strategy is to make a save at any time throughout your game, then proceed to visit each port in what ever order you want. Then when the achievement unlocks, reload your last save so anything that went wrong during your absence doesn’t count.
NOTE: This does not have to be done in a single game as it carries between saves.