I have been browsing musical shops on the harmonica that they sells and I found out they are of different number of notes.

Does it means that it's similar to keyboards that have different keys? And which one should I buy to play?

Those I could find was 16, 20 or 24 notes harmonica.

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My grandfather, who does all his harmonica playing my ear (which bewilders me as much as my music bewilders him!) has a collection of them, all for different things, so perhaps you can gradually build up...!