When I play the cost for Yasuo Windrider to move it to base, does it count as one of three moves I need to activate its effect?

Yes, moving Yasuo, Windrider to your base counts as one of the three movements required to trigger his ability.

According to Rule 407.1, "Moving is the act of a Game Object moving between two Locations on The Board." Since your base is a location on the board, moving Yasuo there constitutes a movement.

As noted in the FAQ regarding Yasuo, Windrider:

• The ability triggers specifically on the third movement in a turn.
• Movements 1 and 2 do not trigger the effect, but they do count toward the total.
• Being "recalled" (bounced to base by a game effect) does not count as a movement, but a voluntary move to your base (such as a Standard Move) does count.