I signed up to HN just to reply. I suggest doing the MNIST dataset. The features are really simple (greyscale pixel values) yet it works very well. You can also play with different classifiers to see how they behave (training time and classification performance).

This will let you setup a whole pipeline from feature selection (in this case just normalise, you can try 0 to 1 or -1 to 1 or subtract mean then divide by stddev, or don't normalize and see what happens) to training the model and evaluating its performance with cross validation. Then you can check your CV results by submitting to the leaderboard.

I took Andrew Ng's ML course then played with the MNIST dataset. I learnt heaps by doing this. Then I got carried away competing in real competitions. :) That's where more advanced feature selection came into play as well as making sure your CV split is representative of the test split.

I was using scikit-learn and just swapping classifiers in and out trying different ones as well as trying different parameters. You can even roll your own logistic regression if you want and see how regularisation affects performance etc.