I'd like to try animating in the following way, but I don't know how to do it or what to watch.
Let's say you have one animation that has two distinct components that you want to blend together, like say a character walking and separately, swinging an object.
How do I create two separate animations in the timeline, then tell blender to apply the transformations associated with both of them but at the same time in one new animation? How does that work?
That is commonly done in the NLA Editor.(NonLinear Animation). If the bones/objects don't have same keyframe then blending should be easy. If the they have matching keyframed pieces like both move the hip bone for example then by default the higher track replaces the lower track. In the N-panel(AKA the sidebar), you can change the blending to something other then replace. I've only used the combine. You can also use the animation influence on each track to determine the amount of influence each track has on a rig/object. Just be aware that that affect all keyframes not just the ones that are shared.