I don't know if it qualify as home experiment, but you can use the internet to get access to thousands of kilometres of optical fibres for free. It allows you to measure a lower bound for the speed of light in the fibres, which is $c/n$, where $n$ is the refractive index of glass, typically around 1.5. This corresponds to $2\times 10^8 \text{m/s}$. Using ping, you measure a round trip time, that is it should correspond to 100 km/ms of round trip.
From Paris, I ping the website of Columbia, in New-York, I have
fred@sanduleak2:~$ ping www.columbia.edu
PING www.columbia.akadns.net (128.59.48.24) 56(84) bytes of data.
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=1 ttl=113 time=125 ms
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=2 ttl=113 time=116 ms
....
64 bytes from www-csm.cc.columbia.edu (128.59.48.24): icmp_req=16 ttl=113 time=112 ms
^C
--- www.columbia.akadns.net ping statistics ---
17 packets transmitted, 16 received, 5% packet loss, time 16023ms
rtt min/avg/max/mdev = 108.585/118.151/132.156/7.728 ms
The minimum round trip time is 108 ms, which would correspond to 10,800 km instead of 5839 km. Off by a factor of 2, but the correct order of magnitude, due to delays in switches etc., which is why we said this is a lower bound.
If one looks more precisely the trajectory of my packets to New York with tracepath
fred@sanduleak2:~$ tracepath www.columbia.edu
1: sanduleak2 0.266ms pmtu 1500
....
3: pioneer.ens-cachan.fr 1.072ms
....
6: vl172-orsay-rtr-021.noc.renater.fr 28.747ms asymm 9
7: te0-1-0-5-paris1-rtr-001.noc.renater.fr 20.931ms
8: renater.rt1.par.fr.geant2.net 30.307ms asymm 9
9: so-3-0-0.rt1.lon.uk.geant2.net 33.780ms asymm 10
10: so-2-0-0.rt1.ams.nl.geant2.net 36.570ms asymm 11
11: xe-2-3-0.102.rtr.newy32aoa.net.internet2.edu 127.394ms asymm 12
12: nyc-7600-internet2-newy.nysernet.net 128.238ms
13: columbia.nyc-7600.nysernet.net 135.948ms
14: ....
We see that the packets travel around (Paris, London, Amsterdam) and cross the Atlantic between Amsterdam (10) and New-York (11) in 127-37=90 ms (roundtrip). This still gives us a 9000 km distance, way too long. I don't know if it is due to the cable trajectory, electronic delays, to small sampling by tracepath or an error on my calculation.
Related to this ping delay, you have the funny 500 miles bug.
Another in-the-lab experiment using cheap material and computers is in the arXiv paper speed of light measurement using ping. However, their measurement is indirect (they measure the propagation inside CAT5 cables), but it should also be doable with optical fibres.
Edited to add: My idea of using tracepath probably comes from Measuring the Earth with Traceroute. In this paper they are more lucky than I was (only 20% slower, instead of 100% !)
First of all, Marek is right that a surface tension exists only between two different materials (well, I would say between two different phases - for example water and ice). So let's rephrase the question as "Are there two phases with zero surface tension?" and elaborate a little on the answer.
The surface tension is the excess free energy (technically the 'grand potential') associated with the area of the interface between two phases. If the surface tension is positive (it always is), the system minimizes the free energy cost by minimizing the area of contact. This leads, for example, to the spherical shape of a water droplet in coexistence with water vapor. But if the surface tension were zero, we could deform the shape of the water droplet arbitrarily, with no free energy cost, so long as we didn't change its volume! We could even deform it to the point where, from the macroscopic point of view, the water and the water vapor seem to be perfectly mixed. But this actually contradicts the fact that there was phase separation in the first place, because phase separation is an indication that a uniform (single-phase) system can lower its total free energy by splitting into two phases.
So zero surface tension between two phases would be quite an unrealistic situation. There are some situations in physics which are somewhat like having zero surface tension. If you look at the surface tension between water and water vapor as the critical point is approached, the surface tension vanishes. But at the critical point itself there is no distinction between the two phases (and therefore no interface), so we can't say that there is zero surface tension. Another situation where it is tempting to say that surface tension has been made to vanish is when oil and water (which don't mix) are emulsified by adding a surfactant or emulsifier which lowers the surface tension. Eventually, rather than two coexisting phases, we have a single phase with a lot of internal structure (little droplets of oil in water or vice versa). But that is a single phase, not two coexisting phases!
So I don't think that any two coexisting phases have zero surface tension. In particular, liquid helium has a positive surface tension (with air I mean).
Best Answer
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.