Purchase a new stereo. Make sure to get one that will fit properly into the slot for your old one. Additionally, when buying a new car stereo you may be able to get a wiring harness that is specific to both your car's make and model AND the new stereo. Buy one of these if you can because it will make the wiring a snap, literally!
Collect these tools and materials: screwdrivers, wire strippers, wire cutters and electrical tape.
You will also need a guide to removing the dash panel of your car that shows how to properly remove your old radio.
Disconnect the ground (negative) cable from your battery. NEVER work on the electrical system of your car with this connected.
Remove the dash panel that covers your stereo. Unscrew and slide out your old head unit. There will either be one single plug with lots of wires that connects to the back of it or a bunch of individual wires. If there are a bunch of single wires going into the back of the stereo do not detach them yet since you will want to do them one at a time to avoid confusion.
Plug the connector into your new wiring harness, if your were able to get a wiring harness for your car. This will then plug right into the back of your new stereo. If you did this you can skip the next section.
Connect each wire yourself if your were unable to obtain a wiring harness to install your car stereo.
Connect the positive wire from each speaker to the positive connector on the back of the stereo first. Then connect the negative wire from each speaker to the negative connector on the back of the stereo. If there is only one wire coming from the speakers, the speakers are grounded at the chassis. In this case you will need to install new speaker wiring.
Connect the ground (-/black) wire of the stereo to a screw or bolt near where the radio is mounted in the dash. Make sure the wire is attached to metal, not plastic or fibreglass.
Connect the 12 constant and ignition power wires (both +/red) to the new stereo. Refer to your stereo's installation manual to make sure these go to the correct places.
Now you should have a nice single connector that hooks to all the proper wires in your car--either from the wiring harness you purchased or the one that came with your stereo that you have now connected all the proper wires to.
Slide your new stereo into the mounting bracket that came with it. In some rare cases you may have to use the bracket from your old car stereo.
Plug the harness connector into the back of your new stereo. There should be only one lone wire left from your old stereo now. This is the antenna. Plug it into your new unit. There may also be a power antenna wire which you will need to connect.
Slide your new car stereo into the slot in the dash. Don't screw it in yet; you'll need to test it first.
Reconnect the negative cable of your car to the battery.
Turn your car on and power up your new stereo. Put some music on and listen to all your speakers to make sure everything works.
Screw your new car stereo into place and put the dash cover back on.