The standard thermostat operates on a 24 volt system. If the thermostat you are changing out uses a different voltage, you should get the appropriate thermostat. The first step, as with anything electrical, is to turn off the power. The next step is to go back to the thermostat.
Remove the faceplate, which exposes the wiring and the screws. Please note, the wires are each different colors and go to a screw with a letter corresponding to the color (red goes to a screw labeled R, green goes to a screw labeled G). Get a piece of paper, and make a drawing of the wires and their destinations, or label the wires with labels supplied from the new thermostat. Unscrew the wires, and then unscrew the base plate and remove it from the wall. There should now be five or six wires hanging through a small hole in the wall. Don't lose those wires by letting them fall back in the hole in the wall.
Take the new base plate, feed the wires through it, and attach it to the wall with the supplied screws. Use your drawing (or the labels) and attach the wires to the appropriate screws. Before you put the cover back on, test the system. Turn the power back on and crank the heat, or AC, up, and see if the system works. If not, double check the wire connections to the screw to make sure there is a metal to metal connection, and no insulation between the two. If it still doesn't work, the transformer may be a problem. However, this is a simple system and the chances are good your new thermostat will work without a hitch. Put the cover back on and adjust to the appropriate temperature.