Would it be possible to make a custom zoomable world map (kinda like google maps), where you can have all the cities of the world but are able to zoom out to the whole world (and no longer see the cities). What programs could I use to make such a map?
AlexP is right of course, you should ask on a geography stack... but that can be daunting/overkill.
The main thing to understand is that worldmaps have too much detail to load at the same time. So they (practically?) all divide the world in "tiles" of increasing resolution --- say the world in a Mercator projection as 1x1 tile (so about 40K x 20K km), then 2x2 (each a quadrant/half-hemisphere), 4x4, 8x8, ... , 1024x1024 (still 4x2 km), ... up to whatever resolution. You can change coordinate systems but you will want to stay with the default/ what people are used to. [In reality of course it's square tiles.] You can choose to limit movement or instead scroll around the world infinitely East-West (not North-South across poles, typically).
The way it works is you get a few tiles (say 2x3, or 12x8) fully displayed on your map plus partial ones on the edges, and it dynamically loads in tiles in the direction you are scrolling. If you zoom in/out one or more steps, it hasn't pre-loaded any so may hesitate a second.
Now the interesting thing are the map suppliers: Openstreetmap is well-known, but e.g. Stamen makes striking ones ("toner" stark black/white maps, my favourite "watercolor" one). Keep in mind there's local ones (say, cycle paths of Switzerland) you can use globally and not see much in most places. The thing is that maps are many layers: a geographical one (e.g. satellite-based suggesting vegetation and height, or political boundaries), one of name-labels (in local language/script, or French, English, ... ), zero or more of "fixed" features (roads typically, or buoys on a seamap; landmarks on a landmarks map), zero or more of "data" features (current weather, local voting results as a choropleth, GDP by county or country, ... ).
You can make your own "tileset" of a fantasy planet, or somewhat abuse it to display a gigapixel image of an artwork you have [see 'non-geographical map']. I vaguely remember a tutorial of taking a 19th century map (of I think Buenos Aires) and generating the tileset; there's also providers like Arcanum.com or the National Library of Scotland who have done the work with historical maps.
If you don't really want a full-blown GIS suite, there are many free web-based applications more-or-less suitable for the creators of fantasy worlds. One of the most commonly used is Azgaar's Fantasy Map Generator a.k.a. Azgaar's FMG. It is very much simpler than an actual GIS suite, but gets most of the job done quite nicely.
Google Maps at least used to have a feature that allowed you to use their entire feature set with your own geographic database. I recall this was actually done for the MMO game There (for their online map of There's world); this started no later than around 2005 (might have been earlier; There opened in 2003).