Step 1: Assemble the parts and tools
Next thing is to find some hanging wire. If you are working with larger stuff, you can use welding rods, which come in 1/16" sizes and have nice, copper coating. For smaller items, I used piano wire, which I got at a shop that sells remote-control airplanes. This comes in very precise diameters... I started with 0.020" and got a range of sizes up to about 0.1"... Whatever you use needs to be stiff and springy, so steel is ideal. Get a lot more than you think you might need... I probably used 40' of it for the mobile shown here.
Then, you need some tools, specifically, pliars and nippers. What you really need are some round-jaw pliars, shown here... these allow you to make circular bends in wire. Crafts stores that cater to jewelry makers will have little ones, which are good to start with, but they can't do much with bigger wire. A google search found me a tool store selling bigger ones, I think they were 6.5"... I'd like a bigger pair still, but I haven't found them. You also need wire cutters; a fairly large pair will save your hands some trouble since steel is tougher to cut than the copper that wire cutters are designed for. A large pair of needle-nose will help with some bending steps.
Wherever you work, do your best to get a hook in the ceiling. Mobiles don't really sit well; they only hold together when hanging, and it is impossible to see how it will look unless it is suspended.
Finally, I want to acknowledge a video that I watched while working on this. The movie is "How to make a mobile" by Timothy Rose, sold here:
I have no affiliation, and frankly, he made it look too easy and I wasn't able to do nearly as well as he did... but, if you want to see how to nip out an incredible mobile in a few minutes, it's worth watching.
Step 2: Start building
Take a short piece of the thinnest wire and bend a complete loop in the middle... then bend a loop in each end. The center loop is the top of the piece, and the end loops should be in the same plane as it. They should end with the open part facing up. Hang 2 of your pieces from them, then hang from the hook to see how it balances.
Next, take a piece of wire and bend loops in either end. Hang the 2-object piece from one end, then add a single object on the other end. Find where it balances and bend a loop at that point. It will take some practice to get this loop to be on the top side of the wire and with the top in the correct position. I find it helpful to bend it completely around one point without moving the pliars... this at least gets the loop on the correct side.