So I have a PC and a Mac and I want to work in between them. Will putting my catalog (or starting a new one) on an external drive and swapping between the two be enough, or are there extra steps?
There are extra steps. Each time you connect to a different computer, you will have to re-link the photos within Lightroom, because the Mac name for the drive will be different than the Windows name of the drive. See these instructions to re-link (particularly Figure 4 and associated text) Adobe Lightroom - Find moved or missing files and folders
You can make the file paths relative (you would have to do this one time only) by moving (using your operating system) all the photos to subfolders under the same folder that your catalog file is located in. To do this, you'd have to follow the above instructions to re-link in Lightroom, once.