Becoming a better researcher is surely one way; when you move beyond searching on ancestry.com or familysearch.org and begin looking in archives, requesting microfilms from the Family History Library in Salt Lake City, looking through historic newspaper files, you've moved past beginner status.
Becoming a better interpreter or storyteller is another way. Unless we come up with an ancestor's handwritten journal (surely the Holy Grail of genealogy) or a detailed newspaper article from the time, we are always looking at a few bare facts and weaving stories out of them. We leave beginner status behind when we leave the realm of fantasy and begin to look at context, history, social practices, and so on, to help us tell our tales.
Birth record of Maximilian Joseph Langer
Becoming a conscientious documenter of sources is a third way, perhaps the most significant way. When we make our first stabs at creating a family tree, many of us ( = me, for sure) fall victim to the siren song of others' family trees, appropriating long and complex branches out of sheer excitement -- "Wow! Look what I found!" But when we begin to discover that what we've copied just doesn't seem right (e.g. date conflicts), or it's incomplete or just plain wrong, we begin to realize that we have to do our own research, make sure our family trees are right and are carefully documented.
Andreas Berneburg Death Certificate
There really are two parts to documenting your sources. The first is to write down where you found the information. This might sound simplistic, but I think we've all experienced that unpleasant moment when we think, where the heck did I find that bit of information? One way to prevent this is to become compulsive about using a program like Evernote to jot down notes as you're researching. The second is to record that information in whatever genealogical program you're using, whether online or on your computer (and be sure to always back it up!).
Where do we learn how to properly cite sources? A simple search on "genealogy citing sources" will lead you to a wealth of information; just perusing the links on Cyndi's List will occupy hours of your time. You will find many, many guidelines to how to cite those sources; this one at the Family Search Wiki covers a lot of the circumstances you will encounter. The basic intention is this: provide enough information so that another researcher could find that exact source again.
Information for Harm Jachens, from the Bremen-Lesum Ortsfamilienbuch
What inspired me to write this particular post? True confessions: I've been carefully accumulating information and documentation, but I haven't entered all of it into my family tree -- I'm hoping that I'll be embarrassed enough by this post to get myself moving! It's very tedious, yes, but it's very important and necessary work. I'll get busy and I hope you will too!