The visualization part is a bit tricky since we as humans are limited to 1-3 D graphics. However, I'd still say Iris is one of the most useful toy datasets for looking at classifier behavior (see image below).
(I've implemented this simple function here if you are interested: mlxtend plot_decision_regions.) Other than that, I think that synthetic datasets like "XOR," "half-moons," or concentric circles would be good candidates for evaluating classifier on non-linear problems:
