How would I remove vertices/etc that are inside meshes that I’m combining together? Other than manually removing the vertices, is there an option to delete vertices/edges/faces that are hidden under the “outer shell” of the conjoined meshes? I’ve heard about something called a “boolean addon”(I think…). Is this what I’m looking for?
The Boolean Modifier would be the right tool for this, I guess.
You don’t join the two objects first though. Select one, then add the modifier to it. Now select the second object for the operation, and select ‘Union’ as the operation. Now if desired apply the modifier to have one mesh for further mesh modeling…
Notice that the second object for the operation is still in the scene. Delete or hide it in order to see the result of the boolean union.
That did the trick, thanks! 
Was the boolean modifier recently added? I don’t recall seeing it before in the modifier list. I’ve only heard of boolean addons before.