For some reason, the faucet in my kid’s bathroom stopped working. There was a slight leak around the faucet’s knobs when turned on, but no water would flow out of the spout.

The old Faucet wasn't working any more.
After much investigating, I decided the supply tubes must be clogged some how.

The 'before' picture, under the sink.
(By “much investigating,” I mean I took apart the knobs and fiddled with the shut-off valves!)
We decided to go ahead and replace the faucet. After some more thought, I decided that I’d replace everything from the shut-off valves up, and the drain pipe (above the trap).
We purchase a cheap (but attractive, according to the wife) 4″, chrome, 2-handle, center-set faucet at Home Depot. I took the oportunity while there to go ahead and buy the new shut-off valves, chromed, flexible, copper supply tubes, a basin wrench, a mini pipe cutter, and, finally, some silicon.
When we got home from Home Depot, I began by turning off the main water supply to the house. I warned everyone that we wouldn’t have water for awhile, then got to work.

Basin Wrench: A requirement!
I had already removed the nuts connecting the old supply tubes to the faucet, so I started by removing the nuts holding the old faucet to the sink. The basin wrench came in very handy here. (In fact, this would have been next to impossible without it!)

I wonder if all my pipes have this much sediment built up?
After removing the old faucet fixture, I had room to remove the rest of the pipes, including the shut-off valves. When I pulled the valves out, I noticed a significant amount of sediment build-up inside the supply pipes. I was glad I decided to replace everything. That sediment was probably what clogged up the faucet in the first place.
I then cleaned up the holes in the sink to make ready for the new faucet. I dropped the new faucet into the holes and tightened up the nuts to hold it in place. While under the sink, I went ahead and removed the trap from the drain, and loosened up the old drain pipe and removed it, also. The new faucet fixture came with a new drain pipe, after all.

Wow. Yuck.
I was warned that the flexible copper supply tubes are a pain to work with, and it turns out they are! After bending, unbending, bending, unbending, and then repeating a couple more times, I finally got them shaped the way the needed to be.

The 'after' picture, under the sink.
I used my new mini pipe cutter to clip off the ends, and then attached the supply tubes to the valves! After that, attaching the valves to the pipes, and then some more bending, unbending, and bending some more, and everything was (FINALLY!) in the right place. I tightened down all the nuts to finalize the process.
With everything in place, it was time to test my work. I asked the wife to watch everything carefully while I went downstairs and turned the water back on. Miracle of miracles, nothing leaked! I got it right the first time. With bucket in hand, I went back upstairs to test the faucet itself. After releasing the valves (still no leaks!), I tried the hot and cold knobs and everything worked great.

No leaks! Yay!
The only thing left was the drain. I turned the faucet off, pulled my bucket out from under the sink, and started attaching the new drain pipe. It took a little longer than I thought, but finally everything was in place and tightened. I re-attached the trap and it was time to test again. No leaks again!

After the clean-up, everything looks great!
I attached the plunger control rod and adjusted it to work correctly. Then, for good measure, I tested the plunger and the sink’s over-flow drain. Everything is working great!
All total, I spent about 1 hour at home depot and about 4 hours working on the sink. I spent a little more than $50. I don’t know how much a plumber would have charged, but I’m pretty sure it would have been at least an arm, if not an arm AND a leg!
… and they said home-ownership was hard.

It's me, chest deep under the sink.