Is the wave going downwards first?
That all depends an what you mean by downwards and what type of change you're interested in.
Let's investigate "downwards": The wave solution describes the disturbance of some quantity from its equilibrium solution. It may be a displacement of a section of string from its rest position (say, in a violin or cello or bass). What is "downwards" for the bass? The string is almost vertical.
Or the wave could be a pressure wave in some medium. In that case, molecules are being condensed and rarified along the path of transmission. How is downward defined? Actually, downward is irrelevant.
I believe I understand why you say "downward" and that is because of a routine association between the variable $y$ and the local gravitational pull direction. That is an association that you need to break. The variable $y$ can represent anything, and for the standard wave solution it is merely the disturbance about an equilibrium position.
Now, on to the type of change. You could investigate how the wave disturbance from equilibrium, $y$, changes as you move along the direction of travel of the wave (the $x$ direction) for a given instant of time, $t$, OR you could investigate how $y$ changes for a particular point, $x$, as the time moves forward ("time keeps on slippin' into the future" according to Steve Miller Band).
Let's consider $y=A\sin(kx-\omega t)$ as our starting convention. If we take a time snapshot, start at $x=0$ and slide along the $+x$ direction, the disturbance initially becomes larger in the (pre)defined positive $y$ direction, then it turns around and goes back in the negative $y$ direction. On the other hand, if we choose a particular $x$ location and monitor what happens with time something different will happen. Consider watching $x=0$, our previous starting point. As time progresses, that point will move in the $-y$ direction, reach a maximum negative and turn around and move back toward equilibrium. Again, downward is irrelevant. And the choice of positive $y$ direction is arbitrary.
The importance of the sign of $y$ only shows up when one makes an association with a physical system.