I often open up my programs and don't even think twice about how they help me make my printer work. The recreational 3d printing community is a cool crowd. They are really supportive, and like many in this new generation, they are all about open information. Hiding information behind paywalls is a thing of the past. The information is all out there for us to learn, you just have to know where to look, or how to look. In that spirit, there are many free or open source programs out on the web for the many stages of 3d printing. The main stages for me are design and printing.
For design, I usually use Google's Sketchup, because it is simple and best of all - they have a great free version. It's easy to teach yourself: there are all kinds of free resources out there - like Youtube... If you have any aptitude for software, it's remarkably intuitive. I use this to design or try to make edits to other people's designs. Most of my designs are very loose prototypes with the two neatest ones being a deck box for Magic: The gathering, and storage bins for Ticket to Ride, neither printed or tested yet.
For finding files to print that I don't have to design myself, there are a ton of links. The majority of my prints, so far, have come from makerbot's Thingiverse. I have only posted a couple pictures of prints - the polite etiquette is this: If you use a free file, upload a picture/rating/comments to let the maker and other users know how it worked out. The best thing is that sometimes the prints don't work right on certain machines, or some users don't know how to edit the files to work, so people with the skills will create "remixes" that will work as intended. I have used the remixes of a few files - the Voronoi elephant, for instance, originally had rounded feet and trunk, but the remixer flattened them so it will stand up properly - most important for printing.
Though a quick google search can tell you that there is no shortage of sites with free/open source .stl files (the files my printing software uses), I thought a quick round up of honorable mentions would be nice.
Pinshape has a lot of duplicate stuff that I have seen on Thingiverse.
My Mini Factory has a ton of cool fan art, jewelry, scans of sculptures from the real world. Categories such as Accessibility, Fan Art, Props and Cosplay, and Scan the World. I could definitely get lost looking at all the neat files they have. Not many Board game Organizers
YouMagine has some standard stl files, a few board game organizers - but you have to look for the right terms. Fashion, Toys, the standard 3d garb that on the big sites.
The Guys at 3d Printing For Beginners have compiled a pretty decent list of 3d model repositories, if you are looking for some good resources.