Can I use unsupervised learning followed by supervised learning?
You can definitely try to first cluster your data, and then try to see if the cluster information helps your classification task.

For example if your data looked like this (in 1D):

AA A AA A A      BBB B B B BB BB BB      AA AA A A AAA


then it may be reasonable to run a clustering algorithm on each class, to obtain two different kinds of A, and learn two separate classifiers for A1 and A2, and just drop the cluster distinction for the final output.

Other common unsupervised techniques used include PCA.

As for your football example, the problem is that the unsupervised algorithm does not know what it should be looking for. Instead of learning to separate american football and soccer, it may just as well decide to cluster on international vs. national games. Or Europe vs. U.S.; which may look like it learned about american football and soccer at first, but it put american soccer into the same cluster as american football, and american football teams in Europe into the Europe cluster... because it does not have guidance on what structure you are interested in; and the continents are a valid structure, too!

So usually, I would not blindly assume that unsupervised techniques yield a distrinction that matches your desired result. They can yield any kind of structure, and you will want to carefully inspect what they found before using it. If you use it blindly, make sure you spend enough time on evaluation (e.g. if the clustering improves your classifier performance, then it probably worked as intended ...)