I have a new computer, and need to transfer my old hard drive to it - how do I do this?

I have a new desktop (it’s actually still in the box). I also have a 1 year old 1T SATA hard drive with everything on it. Seems like it’d be easier to just move the old hard drive to the new computer, but I don’t really know how to do that and make the preinstalled hard drive extra space. Do I just plug the old hard drive into the new computer, and turn it on, or is there some more complicated process to all this?