Right click on the key for the part you want ease in/out->interpolation mode
One approximate approach is to break up the change in distance with additional keys. Set the distance of these intermediate keys to approximate an acceleration. It still looks a little "jerky".
Lets say you want to move an object 8 inches over 4 seconds. At 1 second I would put in a distance of 0.5; at two seconds 2.0; at three seconds 4.0; and at four seconds your desired distance of 8.
A second option is to create some "cam" parts to define the motion of your moving parts. You would mate your moving parts to the cam and then drive the angle of the cam with your animation - while keeping the cam hidden.
Attached are two animations showing the "cam" approach.