To make things look three dimensional, you must know where your light source is. The trick is to make some things look closer to the light source and other things look farther away. If every part of the object you're drawing appears to be the same distance from the light source, that's a very two dimensional drawing. As a rule of thumb, anything that's far away will be in shadow. The darker you make something, the farther away it will seem. On the other hand, the closer something is, the more light it will reflect and therefore, the lighter it will seem. Makeup artists know this and that's why they put highlighter( a lighter color than your skin tone) on the tops of your cheekbones and contour(a darker color than your skin tone), in the hollows of your cheekbones because they know that that combination of lightness and darkness will bring your cheekbones out.
Take this rose for example, in particular the petals. The reason it has to much depth is because I made the tops of the petals super light, white in fact, and the underside of the petals black. All the shades in between also contribute to making the rose look three dimensional.
I was afraid to go really extreme with my darks with the watercolor version. I had the idea in my head that watercolor should look delicate and so I deliberately avoided putting any black on it and instead just used a dark purple. Even without using black, though, I still could've made it darker than I did. Anyway, the result is that the watercolor version looks more flat.
Slanted lines also help.
Put shadows under and around things. Take this overhand in my painting, "Couple In Costume At Balboa Park Centennial". Notice how the line where the top meets the front is slanted upwards. That creates the illusion that the top is going away from the viewer, which adds depth.
Notice how I put shadows under this golden retriever's paws going backwards and made the background near him darker than the rest of the canvas. All of his serves to create the illusion that he's standing on a surface and away from the wall. Remember, three dimensional things will always cast shadows unless their is even lighting all around them.
So in summary, to make something look three dimensional, include plenty of contrast, use slanted lines, and paint the shadow the object is casting.
You can see all these tips in action in the video below.
Painter of portraits and wildlife