Use Dry Wall Anchors

Here are some instructions specifically for putting a bracket on your wall using the dry wall anchors and screws.

Steps

Get a measuring tool. Decide where to place your bracket but it is important to measure each hole's distance of your bracket and apply that distance on the wall by using a marker such as a pencil or a pen.
Make holes. Drill holes by using a drill bit as deep as the anchor. Brush off the dusts.
Put the plastic wall anchors. Push them in the wall with your fingers. You might have to push them in to seat them flush against the wall with a plastic screwdriver handle after you push them as far as you can with your fingers.
Then line the bracket hole up with the dry wall anchor and use screwdriver to slowly thread screw into the opening in the middle of the dry wall anchor. Turn the screw until it's firmly holding the bracket against the wall. The anchors are shaped with ridges so they won't pull out easily once you put them in the wall.
Spackle your mistakes. If you make a hole too big with the dry wall anchor and have to start over in a new place you will need to buy a small plastic container of a material called spackling. This is a white pasty type of material that is used to repair small dents, holes, and other minor drywall damage. You put it in the hole with a plastic putty knife, and smooth it down to look more aesthetic.
Finished.

There might be wood behind part of the drywall that some of the bracket holes line up with. That is most likely why an anchor wouldn't insert all the way. In that case, you can just use the screw without the anchor, and it will be even better secured to the wall than if the anchor and screw had to be secured to the drywall alone. Some or maybe none of the other holes may need an anchor, too. You will just have to try it as you drill your holes for the other brackets. I have found that usually for curtain rods and hardware, that you will run across some of the holes that are lined up in front of the wood window header/frame that is behind the drywall, and some fall in the hollow part that has no wood behind it. In order to determine which parts of the wall have no wood window header/frame behind them knock on the wall to hear the more "hollow" sound. This means there are no wood studs or wall mounts and you can put the dry wall anchors directly in here. The anchor gives you support mounted directly to the drywall, which is usually about 1/2" thick.