Here is a nice applet to visualise the motion of gas molecules in a gas. (Click here for the applet’s url)
The speeds of the molecules are faster at higher temperatures. But at any one temperature, the speeds of the molecules are not uniform. In fact, the distribution of speed follows the Maxwell-Boltzman’s distribution (shown in the histogram at the bottom of the applet). The gas molecules are also colour coded according to their speeds in this applet. Did you notice that the speed of the gas molecules are unchanged when colliding with the walls? When they collide with one another, their speeds do get changed as they exchange momentum. But because they are all elastic collisions, the total (and thus average) KE of the gas remains constant.
The Khan-academy has a very neat write-up on the Maxwell-Boltzman’s distribution. Click here.
The following video provides discussion on the root-mean-square square.