A rheologist friend of mine claims there is no such thing as a yield stress. By this he means that the viscosity of the fluid never becomes infinite (or even effectively infinite), it just gets very high so you have to wait a long time to see any flow in response to low stresses.
Leaving this aside, there are lots of household liquids that show non-Newtonian behaviour. My favourite is cornflour mixed into a paste with water. This is a dilatant fluid i.e. it flows like a liquid at low stress, but at high stresses it becomes effectively solid. Experience suggests this is an excellent way of entertaining nephews and nieces for hours (though they will spill the paste everywhere so be prepared for the cleanup operation!).
To make a clear "yield stress" (pace my rheologist friend) fluid take ordinary shampoo and add calcium chloride to it. Table salt will also do but you need more of it. The electrolyte causes the sodium lauryl ether sulphate molecules to organise into macroscopic phases like cubic liquid crystals.
Re your specific questions: I don't think tomato soup is strongly non-Newtonian unless it's thick, though tomato ketchup is strongly shear thinning and also slightly thixotropic.
A good qualitative way to look at shear thinning behaviour is to watch bubbles in the fluid. Typically shear thinning liquids can suspend bubbles for prolonged periods, especially if the bubbles are small.