If you are level 35 and the doctor still does not talk to you, you may need to get the quest first. Go visit a first aid trainer. He/she should have the quest for you to go talk to the doctor. I’m not sure if every first aid trainer will give you the quest. I’ve always gotten it from the trainer in Ironforge. But I expect any trainer will do.
In most cases there isn’t a lot of difference between eating and using a bandage. I was surprised to find that my level 57 warrior hardly ever used bandages as he leveled. He just sat and ate. However, there are times when a bandage can save your life.
You cannot eat whiles swimming. So if you are battling underwater and you need to heal you’ll have to use a spell, potion or a bandage.
You cannot eat during combat. If you have already used a healing potion and still need healing, food will do you no good. But some classes, such as rogues and mages, can stun or otherwise distract a mob long enough to at least partially apply a bandage. Those extra points can save your life. This is one reason to have the most powerful bandages you can have for your level. They will heal more points faster.
Bandages can also be used on others during combat, as long as neither of you are being hit. But again, even a partial bandage can make a difference. Bandages can also be used to heal pets.
I prefer to use food for out of combat healing. It’s just my personal preference. But I always keep at least one stack of bandages on all my characters, just in case.