Free 3D models
You can search for pre-existing models on the Internet to print as they are, or to modify using 3D modeling software.
Free 3D modeling software
Tinkercad - a browser-based 3D design platform, now part of Autodesk (free version available)
Fusion 360 - free CAD + CAM software for hobby, non-commercial design, and manufacturing projects
Blender - open source 3D animation suite. Enable the 3D Printing Toolbox
Sketchup - comes in free or pro versions. Get the Sketchup STL from the Extension Warehouse
List of additional free software packages from 3ders.com
Free Slicer Software
Slicer software prepares files for printing by converting your file into machine-readable code. This software can allow you to estimate the amount of time and material a print will consume.
3D modeling tutorials
How do I make a solid model (Rhino)
How to fix and repair your 3D files (Shapeways)
Tips for designing 3D printed parts (Hubs manufacturing services)
Other resources for 3D modeling and printing (All3dp)
Other resources
3D Printing Price Check - Calculate how much your print would cost at a variety of pay-for-print services
3D Insider’s 3D Printing Guide
3D Printing STL files: A step-by-step guide
3D Printing geometry restrictions
How to optimize your design for FDM 3D Printing