> airports big and small from all around the world
Well, presumably mostly big.
The CIA Facebook shows 13500 airports for the USA alone, and 6 countries with more than 1000 [0]. So, when you count smaller airports (potentially untowered, or unpaved runways), there is way more than what's listed here.
Most countries (ICAO members) maintain an AIP (Aeronautical Information Publication, [1]) which lists the bigger airports in a country (and definitely the international ones with immigration and customs facilities) in part AD (aerodromes) and is often readily downloadable from the pertinent national civil aviation authority [2].
However, information on smaller airfields is often quite hard to come by.
I've flown small airplanes around in southern Africa and Thailand, and relied on websites, books, and aviation clubs to find coordinates for airfields. See eg. the Airfields Directory of Southern Africa [3] or Tom Claytor's list at the bottom of the Thai Flying Club website [4].
[0] https://www.cia.gov/library/publications/the-world-factbook/...
[1] https://en.wikipedia.org/wiki/Aeronautical_Information_Publi...
[2] https://en.wikipedia.org/wiki/List_of_civil_aviation_authori...
[3] http://www.cometaviationsupplies.co.za/index.php?route=produ...
[4] http://www.thaiflyingclub.com
EDIT: typos, clarity