When you rescue a lost relative, they will give you one fourth of a map to a Lost City. Unfortunately, you only have four missing relatives and there are four Lost Cities, so you will need to obtain map pieces to the remaining three cities through other means.
Luckily, the lovely daughters of the Caribbean are available to assist you. After rescuing your lost family, the daughters should now provide you with maps to Lost Cities (along with items, as before). Luckily, the second map pieces always seems to be half the map, so you will only have to get 9 map pieces rather than 12.
Also of note, in my game, all four Lost Cities were in the same general area, in what is now Belize. Specifically, if you go to Google Maps and look for the following, you will find the area where they should be: 16.367251,-88.489546
I cannot ascertain how much of the map must be completed in order to successfully find one of these cities, but I will be working on figuring that out in the future.
Once you find all four of the Lost Cities, this achievement will unlock.
You can add quest locations to your quest tracker. To do this from the sailing screen, simply select the map icon in the top left portion of the screen and choose the appropriate quest. If track is not available, it is because you are trying to select a partial map. Once you have a quest tracked, from the sailing screen, click the magnifying glass in the upper right portion of the screen, which will open a mini-map of sorts. On this mini-map, you will either see a flashing red circle, if your target is nearby, or an arrow pointing in the direction of your target.