Tabletop games tend to take quite a long time (several hours per session usually), can be quite noisy (it's quite hard to be quiet with 4-6 people talking, play-acting and throwing dice) and often involve the use of small, light objects (character sheets, dice, sometimes playing cards etc). Most of the time, several of these things are made of paper.
Therefore, any location you use for games should at least
- Accomodate 4-6 people comfortably with the pieces you use for your game (this means seating, a table, availability of food and drink, restrooms etc)
- It should be a place where your players are comfortable playing an RPG (which may involve acting out things), many people aren’t comfortable doing that sort of thing in public (commenter contribution)
- Be tolerant of you staying for several hours
- Be tolerant of at least a moderate amount of noise
- Not be exposed to the elements (strong wind and/or rain can really ruin a game)
Going through your examples, libraries are usually not a good idea because of the noise. Coffee shops may be suitable if they let you stay for that long and you're seated inside (or you just happen to live in a very accomodating climate). An internet cafe could certainly work. Game shops are frequently recommended because they're used to accomodating gaming groups and often have things like rule books, dice and people knowledgeable about games available if it turns out you need them, but of course any other location that can and will accomodate you would work just as well.