Wood screws directly into a stud are going to be many times stronger then drywall anchors. When you have hit a stud, use a screw. When you are just in the drywall, use a drywall anchor.

Drilling out the strong wood to replace with weak plastic doesn't make any sense.

If you really wanted to just use anchors, they make metal anchors that can be driven into a stud but the description of them is:

No pre-drilling necessary - drills directly into drywall even if you hit a wood stud. Starts easier than other self-drilling drywall anchors.

I emphasized the word even.

_{(source: easyanchors.com)}

Tools

Hammer Drill

The job will be much easier with a hammer drill, or a drill with a hammer option. Though it can be done using a regular drill, given enough time and patience.

Masonry Bit

You'll also want to put a masonry bit in that hammer drill. A carbide bit is the choice of champions.

Method

Squeeze, Push, and Wait

• Put the tip of the bit where you want the hole.
• Squeeze the trigger on the drill motor.
• Push against the back of the drill motor, or grab the other handle if the drill has an optional side grip.
• Wait. Hold that position until you've created a hole the proper depth.

Pull Out

Similar to the Squeeze, Push, and Wait. Except that you pull the bit out of the hole from time to time, to clear any dust and debris from the hole.