Should I change the strut or the whole strut assembly?

If nothing is wrong with the spring, you can change just the inside strut. A coil spring compressor is needed in this case. Otherwise, see if you can get 'quick-struts' a fully complete coil and inside strut assembly. They cost considerably more but, all you do is remove the old one and bolt in the new one. A front end alignment is almost always required after strut replacement.