I have been seeing a lot of confusion in the 3D printing industry from a lot of people. It seems to be that there is some view that the only way to make 3D printable files is via CAD software. This could not be furthest from reality. There is another applicable method, and it is called polygonal modelling.
This is most often used in the movie, gaming and similar industries. Can be used very effectively in 3D printing and has been done so for a while now. It is an east system that can quickly create non-engineering objects. You could use it for example to model a Pokemon and print it out. You could take it into a 3d sculpting software like zBrush use virtual clay to make something with a lot of surface detail very easily.
Polygon modelling is also great for the industrial designer who is exploring forms, and quickly out put it to a STL for printing. Then show it to the boss or client for approval or further iteration.
Jewellery makers who use 3d printers also use Polygon modelling. Allowing them to create some amazing art. From 3d sculptors of animals on a ring to more like geometry. Then they print it out using the applicable printer. Generally, one that is SLA like. Using casting resin.
The biggest issue with polygon modeming is it is not an engineering program. As such it does not do well if you have interconnected parts that must move and work together. Parts that tight tolerances can be problematic in this system. You can easily create 3d printing errors if you are not carful. Like having intersecting shells. A model that is not water tight cannot be printed. Like you would see used game content.
Some of the software I recommend as follows:
Modo, an easy software to learn. Out of the box it supports importing standard CAD/CAM data sets. It can export to most of the popular 3D file formats, including STL.
zBrush, it is a standard piece of software in the 3D graphics industry(CGI). It is your go to software for doing virtual clay. It can model almost anything, from organics, like monsters, humans to hard surfaces like cars, spaceships etc. It has 3D printing support built in. So it is able to get your creation ready to be exported out to your printers host software.
Blender, it is free and open source. With a very active community of users. Its interface is a little clunky compared to some packages, but hey it is free. It too has 3D printing tools available free of charge. It supports a lot of file formats out of the box, including STL. The people in the community are great, and are willing to help anyone who has questions.
One software package I would not recommend is Maya. It is great for animation, set for animation. A place to set up assets for games, and movies. Its modelling tools are a little clunky. Its features are also implemented almost like an afterthought in many cases.
I dare you to try polygon modelling or the very least not assume that the STL file you have downloaded is from a CAD package. It may very well be from one of the many polygon modelling packages.