Taking a proper people portrait is very different from taking pictures of say a picture of family and you in a holiday travel destination. For one, you are not taking a picture of “I was here” in this so-and-so vacation place. Sure those photos are memorable and show a happy family together for an outing in some far flung place. And of course if you must and do show a bit of the location you were at by including bits of background picture of wherever you were having a vacation.
However to get a nice good portrait photography, such backgrounds are more of a distraction unless you can make it part of your photo composition in an artistic way. In any case, less is more for portrait photography.
The idea of portrait photography is to concentrate on the features of a person. One of the good ways to do that is to fill the frame. With this simple rule, you can’t go wrong. Set your aperture wide to your possible widest setting (eg f3.5) in your lens. Prime lens are quite useful here especially if you have an f1.8 lens that can blot out the background.
How to fill a picture frame, there isn’t any fast and hard rule. You may leave some border space around the person head or go in real close and totally fill up the frame without leaving much left in the frame for any background to show through.
Most of the time, you would take the portrait picture from the chest up, and take care the frame do not cut off a person from any of the joints, like the neck, elbows, wrist. Somehow if a frame cuts through a joint creates a jarring picture, so better to cut off from areas of the body that are fixed like the forearm, thigh, shins, etc.
A good tip for nice portrait photography is to have the person stare into the camera. Generally this would be right, though at other times you may break this rule for other effect, like if person is deep in thought, you would rather have the person stare into space, as they were far away in thought.
And finally, the subject should not pose straight and facing dead front centre to the camera like a tree trunk that makes for a rather wooden pose. Have them turn their body, or tilt their head slightly or even have some interesting outburst of an expression to make the portrait photography come out alive in the picture.