My '03 TL-S makes a very faint "whining" noise when accelerating that seems to come from the transmission.
It still shifts fine for the most part (2nd to 3rd is a little rougher than the rest but no slipping) but wondering if the noise is an indicator of anything?

 

If we are having the same noise, mine has been going on for about 6 years. Mechanic told me it's a certain bearing in transmission and not to worry about it. I haven't worried about it....I just keep the fluid clean(I probably change it more often than others - so,easy, just a drain plug).

good luck!

 

Does yours also make a very faint low pitch buzzing sound (almost like a fart) for like 1/2 a second in 1st and 2nd gear but only the first time after starting and driving? After the first time of running through 1st and 2nd gear it doesn't happen again until the next time I start the car. Doesn't affect function. Just thought it was weird sound.

 

As it is a 2000 model, it makes noises. However, not the noise you just mentioned. Not a very gassy car.

 

Next time, try to listen if that 'faint' buzzing noise sounds as if it comes from the front engine bay on the passenger side?
My 03 CL-S automatic makes the same sound after starting out & driving, as you mention, and it is the ABS pump doing some kind of 'cycling', I believe.

 

Yes, that's exactly where it's coming from.

 

Could be many things...passenger side: could be power steering, drivers side: could be the torque converter.

Good thing you are noticing this though, a lot of the time these things progress very slowly so hard to notice without being complacent.