Mac Mini Re-install

It depends upon the model Mini and what OSX version was on the Mac

How to identify Mac mini models

We may be able to help you without reinstalling if you

Run this and post the results here so we know more about your configuration.

EtreCheck

Install or Reinstall OS X from Scratch

Be sure you backup your files to an external drive or second internal drive because the following procedure will remove everything from the hard drive.

Boot to the Recovery HD:

Restart the computer and after the chime press and hold down the COMMAND and R keys until the menu screen appears.

Erase the hard drive:

1. Select Disk Utility from the main menu and click on the Continue button.

2. After DU loads select your startup volume (usually Macintosh HD) from the

left side list. Click on the Erase tab in the DU main window.

3. Set the format type to Mac OS Extended (Journaled.) Optionally, click on

the Security button and set the Zero Data option to one-pass. Click on

the Erase button and wait until the process has completed.

4. Quit DU and return to the main menu.

Reinstall OS X: Select Reinstall OS X and click on the Install button.

Note: You will need an active Internet connection. I suggest using Ethernet if possible

because it is three times faster than wireless.

This should install the version of OS X that you had installed.

You model Mac has internet Recovery from which you can download a new OS even if the hardware was replaced. See:

https://support.apple.com/en-us/HT201314

Just do as Kappy said. Backup your date.

Boot to Recovery

Format the startup disk

Reinstall the OS via Recovery

You will need an external hard drive formatted for the Mac that is connected to the computer. In the /Users/ folder at the root level of the hard drive select the folder, then double-click to open a Finder window. In the folder you will see your Home (user account) folder with your username below the house icon. Select the Home folder's icon and drag it to the external drive's Desktop icon in order to copy the entire folder to the external drive.

After finishing your reinstallation of OS X simply open the /Home/ folder on your external drive to reveal its content. Select all the folders EXCEPT the Desktop and Library folders. Drag the selection into the new system's Home folder. You will be asked if you want to replace the existing content. Click OK.

Removing your photos will not address your slow issues. You have something else going on. Run the EtreCheck program and post here. Also increase your RAM