A whole other world is waiting for the photographer who pays attention to very small details. This style of extreme close-photography is called macro photography.
Flowers and insects are great obvious choices for subjects, but try to be imaginative and come up with alternative small subject matter. The raindrop photo below is an interesting example of something ordinary that becomes much more beautiful when you look at it with fresh eyes.
Get up close, zoom in, lie on your belly and get down on your knees. You might look like a crazy person in public but you'll be amazed at the hidden details you'll find.
It's useful to use a long zoom (telephoto) lens for photographing small details so you can magnify the subject without losing focus. The capability of a telephoto lens to zoom in really closely will allow you to fill the frame with the subject and prevent the eye becoming distracted by other things.
Using a wide aperture (f5.6 or lower) will help to blur the background if that is the creative effect you are going for.
If you want the whole image in focus you'll need to use a much higher aperture between f16 to f22, especially if you use a telephoto lens. This is because higher magnifications create a more shallow depth of field which will lower the overall clarity of the area around the subject, unless you bump up the aperture.