The handwriting recognition database here is fantastic for testing a variety of simple ML models:
http://yann.lecun.com/exdb/mnist/
In our machine learning class, we would use data from the KDD cup for our projects. Why don't you create a submission for old KDD cups and see if your model can do better than random? 1998 is good for logistic and linear regression:
http://www.kdnuggets.com/meetings/kdd98/kdd-cup-98.html
Using the 2007 dataset, you can try out some of the matrix factoring methods that have worked well for the Netflix prize:
http://www.cs.uic.edu/~liub/Netflix-KDD-Cup-2007.html
These should all be relatively small tasks. Learning how to interpret your results and iterate your models to make them better will take longer than understanding the algorithms.