What is the best approach to dealing with a weathered engine bay?

 

Clean it good first by using an engine degreaser and following the directions on the car. You will have to get in there and remove buildups in tight spaces, an old toothbrush works well. Then keep cleaning, for example if the metal parts are tarnished they can be polished. There is good info on Avalons site about detailing.
If by weathered you mean the paint is comming off clean it up first and see how it looks. Not sure where to go from there...professional painting maybe?