First step is always to put numbers on stuff so we know what we're talking about.
Get a container of known volume, for example a bucket. Put it under the shower and use a stopwatch to time how long it takes to fill. This gives you the shower flow, in liters/minute. Say it takes 30 seconds to fill the bucket to the 10 liter line, that's 20l/min.
Next measure the temperature of the water coming out of the shower, and the temperature of cold water. The difference is your Delta T, which means how many degrees the water needs to be heated.
In order to heat one liter of water by 1°C, an energy of 1 kilocalorie (4180 Joules) is required.
So you can calculate the total volume of hot water used: (flow in l/min) * (duration of a shower in minutes).
This gives you the total energy used, in Joules: (Volume in liters) * (Delta T in celsius) * 4180
Divide by 3,600,000 to get the energy use of a shower in kWh.
Check your utility bill for the cost of kWh, multiply, and you get the energy cost of a shower.
Then you can decide whether it's worth your time or not.
Note that the energy cost will be the same no matter what kind of energy is used to heat the water (electric, gas, solar). The energy is always required. The only difference is how much a kWh costs. So if the cost of gas kWh is close to electric kWh, switching to gas won't save you a dime.
The cheapest option for hot water is solar, if you live in a place with plenty of sun.
If you heat the home with gas, the most likely best return on investment would be adding extra insulation, hunting for drafts, etc, to save on heating, which is most likely much more expensive than hot water.
Example:
Flow 20 l/min
Duration 5 min
Volume 100l
Hot 37°C, Cold 13°C, DeltaT 24°C
Energy cost 2.8 kWh
With 0.20/kWh, that's a cost of 0.55 per shower.
Say 4 showers/day, round it to 2€/day.
In this case, a solar water heater that works, say about 6 months a year and costs 500 is profitable. It really depends on the numbers.
In most cases, a thermodynamic water heater is not profitable, due to ridiculous wildly inflated prices (€2000) and low COP, don't expect it to work in winter if it runs on outside air. So the nice thing about this is there are plenty on the used market, from people who got gouged, so you can get a bargain. And then you mount it on the house ventilation exhaust to recycle heat in winter that would otherwise be vented to the outside. And then it will work. But if your ventilation exhaust is in the attic, it may not be entirely practical to install...
Note it makes no difference whether a mixer is used or not, energy-wise. If you heat a smaller amount of water to 60°C and mix it with cold water to get 37°C water, on first order, the energy used is the same as to heat the whole volume of water to 37°C. Of course different boilers can have different efficiencies, so it's not exactly true, but you get the idea. Another difference is how much energy is lost in the hot water that stays inside the pipes after the shower, but that's usually negligible.
