I thought it would be helpful to create a resource page that explains what I use to make my stuff.
I'll add to it as I learn more.

Disclosure: None of the links below are affiliate links. They are just things I need and use.

SketchUp is an easy to use 3D modeling program. There's a version called SketchUp Pro, with additional features, that costs $590.00, and a version for non-commercial use, called SketchUp Make. This one comes without a price tag. I've always been more than happy with the free version.
Here's a tutorial to draw a cube, tetrahedron and octahedron.
  Update: 16/03/2014

Keyframe Animation is an easy to use plugin for SketchUp that lets you easily add movement to objects in your SketchUp model. The plugin costs $39.00 and unfortunately, it can only be used when you're online.
Once you've seen this tutorial, you're ready to go!
  Update: 16/03/2014

Once the frames are generated in SketchUp, I use Photoshop Elements to scale, crop, edit and turn them into an animated gif. Photoshop Elemement, for $68.88, does all this without a problem (but it lacks, unlike the full version, the very handy feature to Load Files into Stack). Sure there are free alternatives who can do all this too, like Gimp.   Update: 16/03/2014

For my augmented reality experiments, I use the free version of Inglobe's AR-media. Here's a quick start guide.   Update: 24/04/2014

Every now and then, I need nesting software. And a free and good nesting program is hard to find. I use nestlib because it's the best I could find.   Update: 26/05/2015

Wolfram Mathematica is a more professional take on producing animated geometric constellations. I used Mathematica to develop a pumping-thing generator. This software is not easy to learn and the version for hobbyists and enthusiasts costs $300.   Update: 25/05/2015

Tinckercad from Autodesk is a free and online 3D modeling program. I prefer working with SketchUp but I use Tinckercad to prepare solids for 3D printing.   Update: 25/05/2015

I've ordered quite a lot of 3D prints from i.materialise, the online 3D printing service from Materialise. I'm happy every time.   Update: 25/05/2015