Using SLR to focus subject and background both?

I recently shifted from point and Shoot camera (Canon A720) to Nikon D3000. I'm still learning to use it… But one difference that quickly noticed is that when using Auto mode, If i take a picture of a person with some scenic background, e.g. Mountains or flowers etc, it keep the person in sharp focus but blur the background… IT completely spoils the picture.Am i supposed to use a different setting to keep both person/subject as well as background both in focus?

Use Aperture Priority (A or Av on your mode dial) select a small aperture (high f number) and the camera will alter the shutter speed to get the correct exposure,

Aperture controls depth of field, the wider the aperture (low f number) and the closer you are to the subject the narrow the depth of field is.

One of the first steps in taking control of your camera.

When deliberately used 'selective focus' (subject sharp but background blurred) can make your subject stand out from the background, as a rule of thumb only landscape photographers want everything, front to back, to be in focus.

To have both in focus you need to have a small aperture (lens opening) It is likely that the default for your camera in Auto is to make the shutter as fast as possible to stop motion. A fast shutter for any given light level means a larger lens opening. You may need to change a setting in the Auto, or switch over to manual and/or make aperture something you chose - you would use f/8 or f/12 or even a bigger number on the bottom for a small opening.

What you are experiencing are a few things

The sensor of you D3000 is over 15 times larger than you old P&S camera
Depending upon the focal length of the lens, the depth of field can vary when using the same lens aperture

Wide angle lenses pull more depth of field at f/8 than a 300 mm telephoto does at the same lens aperture. If you want more depth of field, you need a wide angle lens AND a smaller lens aperture. Wide angle lenses tend to distort a persons face when the lens is too close to the lens.

What that means is that there are NO special "settings" to duplicate how your old P&S camera captured subjects

You will need to spend time with your user manual and using smaller apertures to create more depth of field.

When you are able, take a class in basic photography so you can learn the fundamentals of photography quickly and do not learn bad habits using the "try and fail" method

Shoot in aperture priority and select a large f/number. This is easily achieved when light is strong. If not blur may occur. To prevent blur, make the f/number smaller bit by bit checking if blur is gone while maintaining the background sharp.

Exposure, hyper focal distance, aperture.too many things come into play.
To understand the fundamentals of photography just download Bryan Peterson's 'Understanding Exposure'.It covers the fundamentals of photography comprehensively.
It is also available as e-book.