I would pop the hood before starting it, and then jump out after starting and see if you can isolate the sound, or at least the general area.
If it is the accessory drive belt, try spraying a little water, or if you have belt dressing spray, on it and see if it goes away. Sometimes a belt is just "sticky". Don't use a lubricant though, you need that belt to not slip.
There are also a series of rollers that have bearings in them, you can try spraying a little lube on the ends of these with the belt running and see if it changes. If you have a long metal rod, you can also touch the end of it to non-moving parts and hold it near your ear, and see if it transmits the sound through, like a doctor's stethoscope. Often you can isolate a bearing in one of the rollers using this method to differentiate where the sound is strongest. You can touch the bolt on the end of the pulley as it is not moving.
I can't say about the trip- the fact that the sound goes away is encouraging, but I can't give you a guarantee per se.
If it is the accessory drive belt, and it fails, it will leave you without power steering, or charging system for the battery.
If you have any friends who are handy, you may want to have them listen and give their opinion of the severity.