thermometers , temperature heat 
For a thermometer to work well should its heat capacity be lower, same, or higher than the objects being measured?

Heat capacity is how much energy is needed to increase temperature?
Therefore the heat capacities would have to be the same for both the thermometer and the objects. 
There is an example in my text book about monoatomic gases converting all their energy into kinetic energy. However diatomic and polyatomic gasses need extra heat to convert to kinetic energy.
 A: A thermometer works by having some property that depends on the thermometer's temperature in a well-defined way.  You put the thermometer in contact with the system you actually care about and wait until the thermometer is in, or near, thermal equilibrium with your test system.  Then the thermometer's properties measure its temperature.
But: the energy required to change the temperature of the thermometer must come from (or flow into) the system that you're measuring.  So the thermometer doesn't quite report the temperature of your original system --- it reports a sort of weighted average of the starting temperature of the system and the starting temperature of the thermometer, where the weighting is the heat capacity.  (Note the difference between specific heat capacity, a property of a material, and total heat capacity, which is proportional to mass.)
So: imagine that you've invented a new type of thermometric material, which for some technical reason has to begin your measurement really hot, red-hot, before it comes into equilibrium.  You want to use this device to measure the temperature of a big pot of water that's sitting on your stovetop.  Which would give you a better estimate of the water's original temperature: if your thermometer is a little grain-of-rice bit of material? or if your thermometer occupies as much space in the pot as the water you're trying to measure?  Relate these to heat capacities and you'll have your answer.
