Remove old towel bar:
Use a screw driver to remove the screws that hold the old towel bar in place. This is applicable if the bar is directly screwed to the wall. If mounting brackets have been used, unscrew the screws on the underside of the bracket. This will give off the bar with the bracket still intact to the wall. If you are using a new bar of the same old model, this bracket will cut your work short because you will not have to install a new one. However, if you are going to use a different model, remove the bracket entirely by using screw drive or Allen wrench.
Fill up the screw holes that the last bar had created with joint compound. When it has dried, sand it with a sand paper before painting in the wall color.
Use the template at the back of the new towel bar to mark the points of the screws on the wall with a pencil. If there is no template, measure the distance between the two ends of the bar and translate that difference on to the wall. You can even just hold the bar to the wall to mark the points.
On the marks that you made, drill small pilot holes. These holes should be slightly smaller than the diameter of the screws of the bar that you have. The purpose is to have a snug fit for the screws.
Install bracket and bar:
Tap the bracket into the pilot holes. Use a hammer to install the screws into the right holes. Use a level to ensure that your bracket is even. Attach the bar to the bracket by first installing one end into a bracket, then the second one so that both bars fit inside.