How to change After Effects frame rate so that it doesn't speed up
If I get the question right, what you're wanting to do is add more frames to the footage to make it play smoother. You have to add the frames because your footage was shot at 59.94 frames per second, and you want to double(ish) the number of frames displayed every second to 120. So every frame there has to be an extra 60 (ish) frames to display to make up the shortfall.

Ok so the first thing to do is to make a comp containing the footage and set your comp to 120fps. You do that in the Composition settings dialog. ctrl/cmdk will summon that boi.

So now the composition has two frames for every frame of the source. Being the professional program that it is it will still display your footage at the correct speed, and by default it will do that by displaying every frame twice. This is pretty much the same as playing back your footage at the original fps. We need to make up some frames that don't exist in your footage.

To generate the in-between frames you have to turn on frame blending. This can be done in the layer switches for the layer (Hit f4 to hide/show them). Clicking in the frame blending square once turns on frame-mix blending , clicking again turns on pixel-motion blending . 



There are two options. Cheap and cheerful frame mix basically makes a 50% blend of the previous and the next frame to come up with the new frame. The second mode is the computationally intensive, sometimes brilliant, sometimes very weird pixel motion. It uses all sorts of clever algorithms to create a new frame by looking at the motion in previous and following frames and warping your pixels to match. A bit like you added a morph between successive frames.There are also 3rd party plugins for time-stretching, notably Twixtor.

Now you won't see anything when you turn on frame blending for a layer, because by default it's not shown in the preview you see in the comp window, because it slows rendering down a bit. You can turn it on for the preview by hitting the frame blending button in the timeline window, as seen below. It goes blue when it's on.



You also have to make sure that frame blending is on in your render settings if you want the final result to render out with frame blending.

You should prolly read the documentation, it sounds like you're a bit confused about things like interpret footage and time stretching and so on.