Hello Lizzie,
Some pumps may be noiser than others, but it doesn't usually cause comment so there may be something else wrong with yours, but it is really difficult to be objective without being able to watch and listen to the system.
One of the commonest problems with sub pumps, is if they are allowed to either run dry, or continuing to run when the taps are closed. The motors can over heat and damage their bearings and seals.
There are traditionally two separate ways of controlling the pump, The first uses a microswitch in every tap, which responds when the tap is turned. It only needs one switch to fail, and the pump will continue to run even when the taps are off. Its also a bit of a job to find which one may have failed. The switches can also fail to turn the pump on, at least the faulty switch is easier to identify.
The other system uses a single pressure switch fitted to the cold water supply. These need to be adjusted so the pump switches off when the tap closes and the pump runs for a short while to build the pressure up. If the switch is set too high, the pump will not develop enough pressure and will continue to run, so the switch needs to be adjusted down.
Having said that you have asked if there is a preferred way of commissioning the system.
There are a few points to watch though;
If the pump is left connecetd to the caravan when it is dropped into the water barrel, the air in the pipe may prevent the water from rising through and priming the pump. This will cause the pump to cavitate be noisy and not pump water properly. The solution is to unplug the pump hose from the caravan whilst the pump is dropped into the water.
I also note you have a filter, These should be replaced at last every season, though in the UK they are hardly necessary. Some systems allow you to replace the filter with a blank.
When using the system for the first time, make sure all drain taps and ports are closed. Then open both hot and cold taps and turn the pump on. When you get water flowing constantly from the cold close it, and then wait for the water to fill the hot water tank and then give a constant flow Close the tap, and then go round the caravan and open the other taps to allow the air out of the remaining pipe work.