Welcome to the Mac side of things flavo. A couple questions: Why Yosemite? Why not go straight to El Capitan?
Usually I recommend a clean install, but I just upgraded my old 2007 iMac and 2010 Mac Mini from 10.8 to El Capitan and it went flawlessly, so I think that's an easy, viable option.
Actually, as I type this, I'm going through a very similar thing right now, replacing my HD in iMac with a new SSD. Ever since I updated to El Capitan the HD is working like crazy, I read El Capitan prefers a speedy SSD to do its thing, and I've always wanted to get the fast bootup speed from an SSD. Anyway, I'm doing pretty much what Steven described, but slightly differently. I got one of those USB external cradles and put a fresh SSD in there, now I'm cloning my iMac HD to it and then I'll be adding a Bootcamp partition with Windows, making it a dual system, then I'll make sure it boots up and runs smoothly from the cradle before I go through the actual installation, which is a PITA.
So in your case, you might want to consider hooking up the raw SSD externally and do the fresh install there before actually going to the trouble of installing the drive in the Mac Mini. If there are programs or data on the Mac Mini you want to keep, you can use the "Migration" software that Apple supplies as part of the installation (I think). Or as I mentioned earlier, just clone and then upgrade to El Capitan.
As to cloning, I never trusted "Time Machine" which is the Apple reccommended method for backing up. I prefer a program called "Carbon Copy" which gives you a bootable mirror image of your drive. But as Steven mentioned, "Super Duper" is good for that too.
Good luck with it!