When I try to start my Celica after it's been sitting for about 10 hours, I get an odd noise that
almost sounds like an idling weed eater from what I'm assuming is my rack and pinion because it only happens when I turn the wheel. Any ideas on what this could be?

This sounds like there is air in the system had the same problem a few weeks back if this is the case you should have bubbles in the RES. Let the engine run lock steering full to full four or five times then turn engine off so it can settle. do this until most of the bubbles are gone and then bring for a drive on winding roads. take it easy for day or two then should be grand keep an eye on you fluid levels of being doing this over and over.

Could also be that your power steering belt needs replacing, and it makes the noise until the belt warms up.

>

I think you're right. It takes a minute or so, but the sound goes away.