Hi there! I'm admittedly a novice and am very close to calling a plumber, but I wanted to see if anyone here had any ideas...

I removed an old vanity and replaced it with one that was about 4" taller. The waste pipe that comes out of the wall sits pretty low, and the tailpipe that came with the new faucets didn't quite meet the p-trap. So, I needed something to connect the tailpipe to the p-trap. Also, the tailpipe is 1-1/4", and the p-trap is 1-1/2" (the p-trap needs to be a larger size because it needed to fit the waste pipe).

So...

1) I connected a 1-1/4" to 1-1/2" adapter to the tailpipe. I didn't put any pipe dope, plumbers putty, or tape at this connection -- just the locknut.

2) I cut a 6" piece of 1-1/2" PVC, and wrapped teflon tape around both ends a couple of times.

3) I fitted the pipe into the mouth of the adapter and the p-trap. It fit really snugly -- no discernible movement. Also, about 2" of the tailpipe sits down in the PVC piece that I cut.

I wasn't hopeful that this would work, but I had my fingers crossed. After I hooked everything up, it seemed to work and stay dry for about a week. I'd been keeping a bucket underneath the set-up and checked it periodically. Yesterday, I noticed some water, so I ran the faucet to see what happened. Sure enough, water bubbled up out of the adapter where it connects to the tailpile, and out of the p-trap where the PVC is resting in it.

Clearly I've done something wrong here, but the nice guys at my local big box DIY store haven't been able to help me figure out a solution... I thought about using pipe dope or plumber's putty, but many sites really caution against doing that. The people here seem really knowledgeable, though, so I thought I'd take my chances. Any ideas? Thanks in advance!

The big box guys aren't plumbers so you're getting advice from the wrong place. Here's how I would have done it. I would have purchased a chrome lavatory tailpiece extension, (see image) and cut off what I didn't need. I would also have picked up a 1 1/2 X 1 1/4" nylon compression gasket to fit on the compression nut to replace the 1 1/2 X 1 1/2" gasket that was in there.
It's just that simple. Now everything should be tight. Good luck, Tom