If you want to "get your hands dirty" with statistics, there is an infinite number of ways to mine data, set up hypotheses, experiments, analyze data sets using various styles of analysis (i.e., Bayesian vs. frequentist), etc. However, your question indicates that you're having trouble figuring out where to begin.
Find a topic that interests you, be it sports, politics, science, commerce/economics, behavioral psychology, biology, and so forth. If there are phenomena in said topic that can be quantified, then more likely than not, someone has already dug up massive amounts of data. I highly recommend Nate Silver's blog FiveThirtyEight, (of which I hold no vested interest/position in) where there's a wide variety of topics with small studies and analyses featured. The topics are well-thought out, and the statistics used isn't terribly difficult to understand or grasp for beginners (no offense to Nate Silver). At the very least, you could build upon whatever articles that have been featured, and use the many data references in the articles to do your own analyses or run your own specific tests.
After you have figured out what topic you'd like, specify a particular question you have in said topic- for example, I just thought of "in sports, do high-margin wins correlate to championship titles, or fatigue?"- and then find your data. The resources on the internet are nearly endless, but you must remember that not all data is quality data (i.e., be wary where you find your data, and whether there are any ethical/quality concerns).
Some useful links include DataHub.IO, where you can find (and share!) many free datasets, and Data.gov, a source of all open data that the US GOV shares. If your programming skills are pretty good, I imagine you can also fetch data from the popular social media webpages, i.e. Twitter, Instagram, Facebook, etc.
Don't forget to have some sort of go-to statistical evaluation software. Most (myself included) would recommend the open-software standard, R, but you'd be surprised how far you could go with something like Microsoft Excel, if your data size isn't terribly large and complicated.