Hello and welcome to the world of commercial espresso. With that kind of kit you really need to become a member of TMC.
Depending on how hard your water supply is you can connect the machine without softener, at least to try it out and make surer it's ok. There are many mixed-bed softeners especially for hot drink machines to be had on ebay. Get a new one. You don't need a very large one, 8L is plenty. Be aware that they need regeneration with salt every few months, according to how much water goes through them. In the long run you will be glad you got one. Use pressure flexpipe to connect from the supply to the softner and from the softener to the machine.
Whether you need an additional filter depends on your local water and pipes in your house. Do your perlators (aerators) on the faucets need cleaning every few weeks? If the do you may want a particle filter before the softener. You can either mount the particle filter on the supply line or on the softener.
You can connect the drip-pan waste hose into the waste pipe for the washing machine, but make certain there is no way water can back up to the espresso machine. The waste pipes are best left hanging loose into the syphon (trap), but should be mechanically fixed so they can't slip or be pulled out.
For water supply you can tap into any cold water line. If your wash machine uses cold water you can put a "Y" on the tap and run one line to the washing machine and one to the espresso machine. If you need information on plumbing ask here again and someone will give you some help.
I assume the Rancilio is a 230-240V machine and it should have a good, heavy duty plug and be attached directly to a 20 Amp fused socket that is in good condition. Extension cords are not advisable for anything over 2 kW and your machine uses more than 4 kW. If you want to make it permanent you can wire it directly into the house mains without using a plug and socket. If you're not completely sure about that have a professional do it. We don't want to lose you before you join.
If you have more questions don't hesitate to ask. Good luck and good coffee!