Does a sound wave have to accelerate to get to that speed in air? Since sound is a shockwave with air molecules oscillating in longitudinal motion, I'm wondering what is the scenario when they just started to oscillate. While the amplitude might gradually increase to a certain limit, does it also mean speed of the wave will also gradually increase?
 A: Sound isn't a thing that moves. Air moves. When we say a sound wave moves, what we mean is the place where air is moving changes. Air does accelerate. But the sound wave does not. 
To generate a pulse of sound, you push the air next to you away and then move back to your original position. Typically you do this very quickly. For a low pitch, you might do this in $0.01$ second. 
This doesn't really create wind. It creates a small region right next to you where air molecules have been pushed away, leaving fewer molecules than normal. Pressure is low here. Slightly farther away, there is a region where the molecules have been pushed. They are squeezed closer together than normal. The pressure is high here. 
The molecules in the high pressure region push themselves apart. This fills in the low pressure region, and creates a high pressure region a little farther away. 
The process keeps going. The place where there is high pressure keeps getting farther away at a constant speed. In regions ahead of it and behind it, air is at rest. In the high pressure region, air is accelerating and decelerating. 
A: You ask about what happens when the molecules "first start oscillating", that depends somewhat on the contact area for the generator of said sound wave. The energy transfer will occur at points of contact with the air around the object according to the objects vibrations/movement. Just like making wind with your hand as a fan really. The wave itself does not really change speed aside from the medium it propagates through and the inverse square law, barring interference. 
The sound wave already has energy from whatever is the source of the sound by the time it hits the air. For example cymbals only make noise when given a percussive blow. 
On the other hand, changes in the wave propagate and may / may not infer with each other potentially causing some emergent phenomena. If you find a scenario where any sound wave phenomena has accelerating features riding on the waves acceptable then you are in luck. Look no further than Bohmian mechanics. Folks have been using sound to simulate some of its concepts. See "walking droplets" for an example of features which accelerate, change direction and other fine feats.
You might find something about what you originally suggested in turbulence as well. The conditions for a sonic boom are caused by an inability for a medium to move away quick enough. Turbulence near the bow shock of a mach-speed object can have a bit of variability beyond the usual speed of sound in that medium.  
Here's some reference material on walking droplets and Bohmian mechanics. I think the second link is a bit of a stretch but probably of interest given your question. 
https://www.youtube.com/watch?v=Xm_6Gi4Jqbg
https://arxiv.org/abs/0907.5284
http://hyperphysics.phy-astr.gsu.edu/hbase/Acoustic/invsqs.html
https://phys.org/news/2013-10-strange-behavior-pilot-wave-dynamics-action.html
https://www.youtube.com/watch?v=Hopd-gKB1Xc
Oh, there would also be the case where the medium (air) has a density gradient somewhere.
A: The sound speed does not need to accelerate but the velocity remains constant by the equation $v$=$\sqrt{\Gamma\frac{P}{\rho}}$. So until the Following factors won't change 
1.Temperature 2.Humidity 3.Density of the air that velocity which is of the wave doesn't change but for individual particle it can change due to the different angle in the wave diagram but the velocity of the wave remains constant until these three factors don't change.  
A: Let me explain it somewhat less scientific matter. Imagine you and your friends playing a game. You stand in a lane and your objective is to pass some signal from one end of the lane to the other end. The first person gives signal to the next who recognizes it and pases it on with a little delay which we can call reaction time. The shorter the reaction time the faster the signal gets to the other end.
The reaction time may depend on the person who is receiving/sending the signal but it does not depend on his position in the line. 
The same holds for the air molecules. The reaction time may depend on pressure, tempereture and frequency but it does not depend on the distance from the source of the sound.
However it's a good question because the sound wave transfers energy. Mass is somewhat condensed energy so you could consider the wave having a tiny mass which would have to be accelerated to the speed of sound. I believe that this acceleration happens during the creation of the wave so the wave itself doesn't accelerate.
