Obtain four DPA rubber mounts. These enable you to attach a microphone to your instrument in a safe manner with very minimal risk of damage. DPA mounts come in one size for all string instruments, so violinists, violist and cellist can use the same ones. You should be able to find the mounts either online or in a local music store that carries string instruments and supplies.
Obtain four lavalier microphones with wireless body pack transmitters. Lavaliers, usually referred to as lapel microphones, are commonly used to amplify string instruments and allow for easy movement around the stage if necessary. They are thought to preserve the natural tones of the instruments more so than other types of microphones and are easily found at audio supply stores.
Obtain a public address (PA) system. This system comes with a mixer, amplifier and speakers. Without this important component, you'll have no amplified sound.
Set up and power up the components to the PA system at your performance venue. One common configuration is to have a speaker on each side of the stage.
Attach the lavalier microphones to your instruments using the DPA rubber mounts. The DPA mounts go between the tail piece and bridge of your violin, viola or cello.
Turn the wireless body pack transmitter on. Make sure it is set to the same channel as the mixer, otherwise no sound can be transmitted. The wireless body pack transmitter is only about the size of a deck of cards and can easily fit in your pocket.
Adjust the sound levels on the mixer. Play through a few measures of a piece to get an idea of how the amplification sounds. You may have to make minor adjustments to the direction of your microphone for optimal sound. Once you tweak the sound to your liking, you've successfully amplified your string quartet.