Replacing a mouse has always been one of the easiest things related to a PC. However, now it is no more complicated than changing a light bulb.
If your old mouse uses a serial connector, you have a fairly old PC. You will not have an easy time finding a replacement with that style of connector. However, you can find an adapter (although it is somewhat rare too) that will allow connection of a PS2 connection to a serial port. Your PC probably doesn't have a USB connection, but you can also find an adapter that allows a USB connection to a PS2 port. Therefore, with two adapters, you may be able to connect a new generation mouse to connect to an old PC. We are aware of no technical limitations that would prevent this to work.
If your old mouse has a PS2 connector, and your PC has USB ports, you can choose either a type of connector for your replacement mouse.
Disconnection of PS2 connectors should be done only with the PC turned off. In reality "hot swapping" usually results in no problems, but the potential for serious damage does exist.
USB mice can be connected and disconnected at will. The PC does not need to be turned off to add a USB mouse.
These days, there are no configuration steps to go along with installing a mouse. Some products may offer add-on applications (software) to improve the user experience, but these are not required for the mouse to operate.