Soon I will be replacing my studio computer. Right now, I still have access to all my stuff I had with Platinum because it is still installed on my current setup. When I replace my computer, how do I go about re installing Platinum to retain all my extra stuff?

Just reinstall as you did before. Hopefully you have saved the install files? That is a must. But if you didn't just use Cakewalk Command Center. Simple.
I have saved install files from day one, years ago. IOW, save everything......
 

If you have never deleted the files in the Command center download folder this will save you having to re- download the files again. I makes installing Sonar much faster.  You can copy those files to an external drive and then copy them to your new computer. But there's a method you need to follow. 
COpy the files  and store them.
Now install Command center to the new computer. This will create a new download folder.
Now copy the files to the new location. 
Now when you run the installers for each product CCC will use those files or updated them with new downloads. 
The CCC folder is found in Program Data/Cakewalk/Command center/