There are several methods to get started...but I usually end up spending some time zoomed in very close and working with a very small brush size.
I think the important part is using a non destructive technique...so don't erase the background. Use a layer mask so that you can 'paint' it on or off as you go.
I usually start with either the magnetic lasso, the magic wand or select/color range. It also helps if you use an adjustment layer first, to turn up the contrast between the subject and the background.
Once I have the selection made, I turn it into a layer mask (on a duplicate layer). Then what I like to do, is to create a color fill layer with an opacity of 30-50% and put that between the layer with the mask and the background. Then I just zoom in and go around the outline with the brush. When it comes to hair, I use a small(ish) brush with soft edges to paint in the mask where the background shows through, then come back with a smaller brush that has harder edges and paint back in the stray strands of hair. It helps to play with the opacity of the brush or choose a tone between black and white for your brush.