I want to build a washroom in the basement of my house. How do I begin?

Answer 1

You have to begin by figuring out how you are going to get the drain water out. If there are no drains at the bottom of the house now, it may be that your sewer line is well above the basement floor level and you will have to provide a sump pump arragement to move the water up to sewer level.
If you do have drains (like in the floor or for a washer), then you have to figure out how to get the waste water to them or where to put them to drain the water.
In a plumbing sense, getting water to the wash room is just a matter of running pipes to cut off valves with fastenings for the rest of the stuff (fauscets, machines) you want in the area. In a practical sense, how messy it is will depend on whether the basement is finished out so you have to cut into walls to hide pipes or unfinished so you can run pipes on the surface.
You will have to find a place to cut into the existing cold and hot water and size the pipes for use at the room (not too small.)
You will almost certainly have to have a permit and will at least have enough knowledge so that you don't get sewer gas backing into the house or drain water being sucked into the drinking water pipes.