I downloaded "Slime Rancher" on Steam on a Mac computer. When I try to play it on my new PC, it is very, very slow. My suspicion is that it is because I downloaded a Mac version, and I need to buy the PC version for it to work.
Your suspicion is incorrect.
First: When you purchase a game on Steam, you automatically get access to all available versions of the game. Windows, Mac, and Linux versions of games are not sold separately -- if a game is available for multiple platforms, you will get access to all versions, and you will always download the version for whatever operating system you are running. (There is no way to download a different version, even on purpose.)
Second: Mac games will not run at all on a Windows system. Period. The fact that the game is running is proof positive that you got the right one.
Many Steam games, including Slime Rancher, support Steam Cloud. This will allow you to transfer saved games between computers (which will usually -- but not always! -- include transferring games between Windows and Mac versions). Steam Cloud is automatic; you do not need to take any extra steps to transfer your saved game.
That being said, if the game is running poorly on your PC, there is probably something wrong with the way you have your computer configured. Without any hardware details, though, it's difficult to say what might be wrong.