Here’s one way of doing just that:

  1. Learn the basis of the Blender user interface. There’s a nice introduction at blender.org. The introduction uses the term “window” for the 3D-view and other views. I will call them “views” and refer to the content as “panels”.

    Quick version of the introduction:

    • Right click the borders between views to join and split them
    • Middle-click/drag to move the contents of a view around
    • Scoll the mouse-wheel to zoom in or out from the contents of a view

     

  2. Start out with a 3D-view, the UV/Image view and a buttons view, as shown here:
    start

     

  3. Select the cube, press Tab for edit mode and u to unwrap.
    Select the bottom choice: Unwrap (smart projections).
    (If you have complicated 3D-object you might want to mark seams and unwrap with another method).
    Click ok.

    unwrap

     

  4. In the UV/Image view, select Image->New and click ok.
    Scroll the mousewheel down to zoom out, if needed.
     
  5. Press F6 and select texture type Image.
     
  6. With the tiny button in front of Load, select the
    image you created in step 4 (Untitled)

    texture

     

  7. Press F5 and in Map input click the litte UV-button
    (lower right of the screenshot)

    uv

     

  8. In the 3D-view select Texture paint instead of Edit mode

    texturepaint

     

  9. Press F9 and select a color in the Paint panel

    color

     

  10. Now, just paint directly on the cube! Notice how the image that is the UV-texture also updates when you paint. The cube can be rotated with the middle mousebutton.
     

  11. In the UV/Image view, enable painting. Now you can draw directly in the UV/Image-view too.
    uvpaint

     

  12. It is possible to turn off the grid and axes in the 3D-view with View->View properties.
    grid

     

  13. If you switch the 3D-view back to edit mode with Tab, you can paint in the UV/Image view while also seeing the lines of
    the UV-map. Switch back to Texture Paint with Tab if you want to change the brush or color.

    editmode

     

  14. Press F12 to render.
     
  15. That’s it!
    render

     

Now you know how to paint directly on your 3D-models in Blender. :)