Whinning Noise when cold.

check alt bearing,open hood & listen while running,turn off engine and remove drive belt then start engine to see if noise goes away.

Does the noise change or go away if you turn defrost on-which will engage the compressor-if so may be the clutch dragging.

I would check the idler pulleys on the belt first. Then maybe the alternator. If your battery is getting a little weak it could be working the alternator hard when you first start it up. I don't know how loud it is though. The alternator wouldn't be real noticeable but some people would hear it.

the noise may be because your power steering fluid level is low when cold, as the fluid warms up it expands enough to stop the whine, you should have the leak checked by your favorite mechanic shop

check your idler, and tensioner pully's, i've seen a few trailblazers that whine and it turns out to be an idler pully, also check the power steering fluid and make sure its full in accordance with your owners manual

Your transmission is going to fail sooner or later, the whining noise you hear is the the clutch packs in the transmission going out.

possible power steering pump.

Your transmission is not shifting into overdrive, it is staying in 3rd gear. I have the same problem.

If it's on turning, it sounds like the power steering pump. Could also be your belts (power steering & and/or accessory) need changing