I'm a raw newbie trying to start the companions' quests. I have to give a shield to Aela but I can't find her anywhere. The quest destination marker tells me to go through a door and then immediately to go back through it, and she isn't on either side. Any suggestions where I might find her? Thanks.

I think she's either downstairs Jorrvaskr or in the back at that point.

Yes, most likely in Jorrvaskr's living quarters. If not there, she's probably outside of Whiterun hunting.

She's in the living quarters downstairs. The arrow is pointing the wrong way.