Well, let us assume that we're using roughly 30 gallons of water (probably not a completely filled tub but enough for a bath). Water is cheap it's about $1.50 per 1000 gallons in the US. So, 30/1000 multiplied by 1.5 gives us $0.045 or 4 and a half cents for the water.
From now on i'll be in metric units because they are enormously easier. 30 gallons is approximately 110L. The specific heat of water is about 4180 Joule/(kilogram * degree-Celsius). 1 Liter of water weighs 1 Kilogram. So 110L * 4180J/kg-C gives us 459,800 Joules of energy needed to heat your bath 1 degree.
Now, I would say your baths are probably somewhere around body temp or more so lets use 100 degrees Fahrenheit which is about 38 C which is about 20 C higher than room temperature (65 ish though i doubt your water is that warm coming from the pipes.) so multiply 459,800 by 20 to get 9,196,000 Joules of energy to warm the water in your tub.
A kilowatt hour is equal to 3,600,000 Joules, so 9,196,000/3,600,000 is about 2.5 kW-h of energy. That said, your water heater is only 80-90% efficient. so lets go with 3 kW-h. If you have an electric heater, then you are looking at about 12 cents per kilowatt in the US. If it is a natural gas heater, then it is less (though i can't find a number with my brief search of google).
So there you are, your bath costs about 36 cents for the heat and 5 cents for the water but these are rough estimates, and probably a little low. but if you think of the energy in terms of what else it can do and not just price it is quite astonishing. 3 kW-h can light a 100W lightbulb for 30 hours or power your computer for a day or so. It's a lot of energy that, in my opinion, means more than the fact that it is just 50 cents out of your pocket. and if you are worried about carbon dioxide emissions, then know that the average CO2 emissions per kW-h is 2.3 lb.