TinkerCAD – a web-based program which is very simple to use and features several excellent tutorials for beginners. For additional features and functionality, try its downloadable cousin 123D Design.
Meshmixer – This program is useful for evaluating your designs for stability and adding support structures if needed.
Sculptris – A slightly more artistic approach to 3D modeling. Whereas programs such as TinkerCad use basic geometric shapes as building blocks for your designs, modeling with Sculptris is more akin to shaping a piece of virtual clay. If you have an iPad or Android tablet, you might also be interested in 123D Sculpt, which has a similar functionality with an interface designed for touchscreens.
Free shape depositories
My Mini Factory
Smithsonian X 3D— Iconic Smithsonian collection objects
NASA 3D Resources— 3D models, textures, and images from inside NASA