I dug around and couldn't find my plumber's putty or teflon tape. Long story as to why this is, and I'm not going to go into it. Off to the hardware store. At the hardware store, I started reading the instructions on the plumber's putty packaging and it said not to use on plastic. My new faucet kit is plastic where it meets the sink, so apparently plumber's putty is not the way to go. No wonder the instructions suggested silicone sealant. So I got the silicone sealant.
I got back home and continued installing the new kit. I took it slow, trying not to screw up anything. I got to the last part, which is to connect the p-trap. I connected the p-trap, but it wouldn't tighten around the drain "tailpiece." The p-trap was working fine and it did fit fine on the old drain, so there is no reason I should have to replace it.
I knew the cone taper gasket was loose on the new plastic tailpiece but had hoped that after tightening the nuts, the looseness would be gone. Not so. By loose, I mean that the slip nut and gasket would just fall off of the tail piece due to gravity. I went to the hardware store in search of a smaller gasket. I picked up the smallest size offered for the gasket, took it home, but it was no tighter. LESSON 1: Take that cone taper gasket with you to the store. Try it on the various standard drain tailpieces at the store and you'll find out what size that gasket is.
I went back to the hardware store where I got the faucet kit. If this is an odd size, perhaps it's a common problem and they would have the "fix" or adapter. The guy at the store gave me LESSON 2: there are only TWO sizes for the tailpiece; it's either 1-1/2" or 1-1/4". There are NO other size in a house. My gasket is already for a 1-1/4" tailpiece; they don't make it smaller.
So he opened a new package of the faucet kit I purchased, and I proceeded to show him my problem. Lo and behold, my gasket fits the tailpiece just fine! Apparently I have a defective tailpiece in my kit and I'm going crazy trying to get a gasket to fit it.
Then I ran off to an evening Mass and the store was closed by the time I got back home. So hopefully tomorrow I can just exchange the tailpiece. I don't want to take apart everything and exchange the whole kit ... and have to reinstall and wait for the silicone sealant to cure.
Note to self: top to bottom, the assembly is drain tailpiece, p-trap slip nut, cone taper gasket with taper facing down