I went through exactly that when I replaced my 1 TB (slow) 5400 rpm HDD inside my Mac Mini with a fast Samsung 256 gig SSD. Here are the steps you need to take:
1. Purchase an external case that can accommodate the "largest" of the two drives, in terms of dimensions. There is the distinct possibility that both drives are of the same size (That was the situation for myself. The 1 TB drive went inside a nice, slim external case).
2. Install the SSD inside the external case.
3. Boot up your Mac Mini, and then connect the drive to one of the available ports on the Mac Mini.
4. Use Disk Utility to Erase, Format, and Partition the SSD.
5. If you don't have it, download the "free" version of SuperDuper!. You can get it from here:
http://www.macupdate.com/app/mac/13803/superdupe
Don't worry about the "price", as you can use it in "free/demo" mode. However, you should start getting into the habit of doing disk maintenance/repairs/back ups, and a purchase of SuperDuper! (assuming it is for your back up needs) would be a wise choice.
6. Install SuperDuper!, then have it back up/clone whatever you want on your 500 Gig HDD onto the 500 Gig SSD (if you made two or more partitions on the SSD, then you would need to pick the one you need for the cloning based on size). The backup/clone needs to include, at a minimum, all the "pieces" for the OS. Hopefully, you have at least the OS (from which you boot your Mac Mini) on either a separate partition, or everything is on just 1 partition. This will create a
bootable backup.
7. To test this, restart your Mac Mini, but hold down the Option key. You should then see a screen that has a "list" of startup disks. Most likely, you'll see both the one for your 500 gig SSD and the one for your 500 Gig HDD. Click on the one for the 500 gig SDD, and then click restart. The machine should start up from that.
8. Shut down your Mac Mini, unplug it, and "swap" the 500 gig HDD for the 500 gig SSD (that is remove the HDD, and install the SSD). Then, install the 500 gig HDD inside the external case. You will then be good to go, and can do whatever you want with the 500 gig HDD that is now an external drive.
9. You did not say which OS you are using, but assuming it is 10.7 or later, the only minor "issue" with this is that SuperDuper! does not backup/clone the Recovery Partition currently on your 500 gig HDD. But, there are a couple of simple ways to re-create it on the SSD. Let us know if you need help with that.
