You are right those are somes of the "rules" of four-part writing. But normally those rules are just the result of a specific voice movement that consist on moving internal voices as little as possible while avoiding paralell 8th and 5ths.
In four-part writing you would not normally have the same C note in the bass for 3 measures or jump around with the internal voices so much.
In my opinion, you can jump directly to learning four-part writing without counterpoint. That said, it is probably better to do both at the same time: four-part writing and counterpoint.
The "practical manual for harmony" by Korsakov teaches, step by step, exactly what you are trying to do. I think it is really useful for learning four-part writing. But I am sure there are other good books for learning four-part writing.
For counterpoint, I used the Kent Kennan book that @PeterthePapercomPoser recommended and it was really useful. I think, even just learning two voice counterpoint, will help you a lot understand harmony much better!