Answer
The yo-yo will roll towards the right, i.e. the direction in which the string is pulled.
Work Step by Step
We first assume that the yo-yo will roll smoothly. Thus we can view the motion of the yo-yo as a pure rotation about point $P$, which is the point of the yo-yo that touches the surface.
We know that this point is always instantaneously stationary. As the force $F_1$ acts horizontally it will result in a torque $\tau$ about $P$ that accelerates the yo-yo in the clockwise direction about that point. If the yo-yo rotates in the clockwise direction, it will move to the right.
Now, what would happen, if our assumption, that the yo-yo will roll smoothly, were wrong? This would mean, that it would slide over the surface. With $F_1$ then being the only horizontally acting force on the yo-yo, this would result in an acceleration (and thus overall movement) towards the right as well.
This is all we need to know to answer the question.
In fact, $F_1$ results in a certain static frictional force at $P$, directed towards the left (as the point would otherwise skid towards the right). As long as the force $F_1$ is weak enough, so that the resulting frictional force does not exceed its maximum value (depending on the coefficient of static friction and the normal force of the surface on the yo-yo, i.e. the mass of the yo-yo), the yo-yo will roll smoothly along the surface.
