A faulty Thermostat can be diagnosed by 2 very distinct issues.
The engine will overheat rapidly, and the air vents will only blow cold air (where the thermostat has stuck closed).
The engine will never get up to temperature, you will see normal operating temperature at about 40-60 degrees instead of 80-90. The air from the vent will be luke warm (where the thermostat has stuck open).

This change was on a mk5 fortwo using part number Q0003179V009. You will also need 3 hose clips, a flat blade screwdriver and an E12 socket.

First you will need to drop the engine a little just to give yourself some extra space. Follow the first steps in this guide to do this.

You'll see the thermostat located here:

Remove the vacuum pipe from the FPR, and pull the breather pipe out of the way.

Unplug the temperature sensor connector.

Remove the hose clips. Note, the standard smart clips are single use and are crimped on.
Use the screwdriver to gently prise the standard clips open. There are 3 pipes on this model.

Now remove the 2x torx bolts (using an E12 bit) holding the thermostat onto the face of the cylinder head.
They can be quite tight to get to, but if you move the transfer pipe bracket it gives you a little more space.

Now pull all the water transfer pipes off (remember of course they will be full of water!)

Now remove the old thermostat. You may need to rotate it a bit to be able to remove it easily.

Clean the mounting face on the block.

Feed in the replacement thermostat, and connect the pipes! You will struggle to tighten the hose clips if you leave this stage to later.

Now seat the thermostat onto the block face. It should locate with the seal - use one of the bolts to locate if it helps, then tighten down.

Remember to reconnect the sensor.

Reconnect the breather hose and vacuum pipe.

Finally, remember to raise the engine back up again!