Basic first aid:
It is imperative that you start learning basic first aid. Being able to assess the type and severity of a wound, illness, or condition is vital. Start by checking out this link, which is a good beginner's guide to help you learn what to look for. Without an understanding of how the human body works, you will not be an effective provider of first aid. Therefore I strongly suggest that you study the basic functions of the human body at this link. If you do not have an obvious cause of injury, suppose that you come upon an unconscious person along the road, use the ABC's--that is, make sure they have a clear Airway, that they are Breathing, and check for Circulation, by feeling for a pulse (and its strength) at the major sites.
More likely in the dangerous place the world has become, you will be dealing with wounds. In general, wounds should be cleaned very well, kept moist, and kept protected. So , make yourself some saline (a rounded tablespoon of sea salt (no added metals, PURE salt) to one liter of distilled water makes .9% saline, or "normal" medical saline) for irrigation, clean the wound, pat it dry, and put some antibiotic ointment on it before placing a nonrestrictive dressing on it. Mind you, this does not apply to Zombie bites. For those, the only option is amputation, or a bullet. For a very good basic look at wound care, check out the CDC's disaster guide to wound management here. If you are desperate, you can look here to get the general idea how to stitch a wound. I warn you, it is basic, but then, so is sewing your skin back together.
Make sure that you get as many medical supplies as possible. For this section, I am leaving out medicines. That being said, my suggestion to you is to completely ransack any medicine cabinets you find, and sort the contents out later. With pharmacies, hospitals, doctor's offices and the like you can be more selective, after you know what you are looking for. So, make sure to grab: Band-aids, curlex, various sterile pads (4x4's, etc), abdominal pads, maxi pads (these are very versatile, useful for heavily oozing wounds), air braces for arms and legs (can help control bleeding when you inflate them and be, you know, braces), as many ace wraps as you can find (work well over dressings when you have no tape), natural tears eye drops, antibiotic ointment, sugar, vaseline (sugar and vaseline can be mixed together to make a a paste to treat pressure sores), burn cream, manual blood pressure cuff, stethoscope, oximeter (and batteries for it), ice and heat packs, iodaform gauze for small wound/nose packing, LOTS of gauze, both in rolls and in pad form, some non-stick coated gauze pads for burns, steri-strips (which help hold a wound closed), rolls of silk tape, paper tape, needles, silk thread, dental floss, alcohol, hydrogen peroxide, surgical supplies such as scissors, scalpels, and forceps.
My main suggestion here is to first get a recent pill guide. It is VITAL that you understand what you may put in your body, and what it might do, especially if you have medicine allergies. Second, you need to make sure you understand what some human averages are pertaining to vital signs, so you don't accidentally kill yourself or someone else by giving a blood pressure med to someone who doesn't need it. So, another list: Acetaminophen, Ibuprofen, diphenhydramine (for allergies), meds for blood pressure (such as clonidine), any and all diarrhea meds (this will dehydrate you fast if left untreated), milk of magnesia for indigestion and constipation (which can rupture your insides if it gets too bad, so I also recommend trying to find enemas and suppositories to break up impactions, aka "big hard poop"), as many antibiotics as possible (make sure you aren't allergic!).
You may want to get some pain meds, but be extremely cautious when doing so, because most of them can be addictive, and the last thing you want is to get your neck ripped out because you were so high you thought the zombie was John Lennon or something. It probably isn't a bad idea to get some syringes and anesthetics for sutures and whatnot, but please, PLEASE read my next (and last) bit of advice first.
READ BOOKS! Find as many books on the diagnosis of disease, emergency medicine, herbalism (nice link here), first aid, and the like as you can. Knowledge is the key to survival, and raiding a medical library is a good way to keep yourself alive and healthy. Not to mention you can train and teach others, and if society ever starts to recover, maybe become a valuable commodity to your new community.
OK, all done. Sorry this was so long, but this one I really wanted to cover a lot in, because this is probably the most important lesson for long term survival we can learn. Thank you, mom, for the help in creating and organizing this list, and for raising me to love medicine. It sure is coming in handy.