Hospital is your best friend DO NOT ever do anything to hurt what you are doing at the hospital. Do not steal anything from the Hospital unless you are in desperate and I do mean Desperate straits (in this case, stealing should only be done behind closed doors to avoid hostility). The most effective way to use the hospital is to send your wounded and sick characters to the hospital on runs. Don't use resources on them that you can get them for free! Your survivor must be at least WOUNDED or SICK, as they won't do anything for you if you are only a little banged up (they will, however, bandage you and put you on meds in the case that you are at least sick and slightly wounded, or wounded and slightly sick, which is the best person to go for the free treatment). This will reduce you to slightly wounded, or slightly sick. Usually Bed rest at that point will get you better. Or in the case of slightly sick, herbal medicines. There is an npc walking around with a med bag (square white box with red cross) Click on that and nurse will bandage you up. or give you meds. OR both. You can feel loot the hospital in any of the rubble piles and some times you can can even find meds there. I would not trade these with the doctor, as he has only basic items to give you for them. You are far better off trading them with the guy who shows up at your house every third day, or the guy at the garage whose dad needs meds. On your second or third trip to the Hospital, there will be an event where shelling has collapsed parts of the hospital. You can clear the rubble in those locations and you will almost always find medical supplies. Again I wouldn't just give them to the hospital. You might get a morale boost by doing that. But unless you have guys that are depressed or broken, you are far better off keeping meds and trading them. Giving 2 meds to the kids who come and ask for them, will greatly improve the morale of everyone in the house.
The doctor's trading inventory is behind where he is usually sitting. It is fairly easy to steal from it with a lockpick; however, Stealing anything from the hospital will caused negative mood set to most survivor.