What is the best way to make smoother logo animations?

"Smoother" is a vague term but I can give you a couple ideas. First, use the "Easy Ease" F9 key on your keyframes to have objects slow to a stop. This affects keyframe interpolation (the placement of the steps for each frame between a start and end point.) You can slow in and out then and not be stuck with all linear movement that jerks to a stop or start. Secondly, if you are animating a chain, like an arm, try keyframing the various joints sequentially rather than at the same time. For example, if you are trying to make a character pound a table, start with the shoulder joint, then a few frames later animate the elbow, then a few frames later animate the wrist. So as an arm starts to swing down the wrist actually rotates back and then catches up later on impact. You will see smoother movement when various components aren't all keyframed at the same time.

I use a third party plugin called Motion by Mt. Mograph for After Effects keyframe refinement. I find that highlighting the keyframes, then keeping dragging the middle bar for speed to around 65-70 works really well for me. This will easy ease they keyframes as well. I also will view the motion graph and drag the bell curve on both sides so it is an even bell curve. I do relatively simple logo animations daily for my full time job and this simplifies my workflow. I also agree with Erik on timing.