Does it count as one of your plays for your turn to do this thing?

A: No. That’s the whole point of it being “free”. It does not use one of your plays. Depending on the game we’re talking about (there are cards like this in Chrononauts and Back To The Future, in addition to many in Fluxx editions), you might only be getting one play per turn, and whatever this thing does won’t use up your play for the turn.