Welcome to the world of outdoor fun!

I think the suggestion to stay near your car for the first couple of trips is good advice. Staying near the vehicle will allow you to have extra gear handy and give you the option of a swift exit, should the need arise.
As for specific gear, here are the essentials I always have with me:
- first aid kit. The basic contents are covered well in the posts above. You can never have too much antibiotic ointment or bandages.
- extra clothes. I pack with the current season in mind while considering the possibility of inclement weather. I always have a rain coat (mine is a thin, Gore-tex lined one) and water-resistant, nylon pants, as well as a thin thermal shirt and pants for layers (these must not be cotton). I also like to have a pair of leather or wool gloves and a wool or acrylic beanie/stocking cap, in case it becomes unexpectedly cold. Finally, and I can't emphasize enough, good footwear is essential. I prefer high leather boots and wool or wool-blend socks. Take lots of socks if you will be around water or doing any hiking.
-signaling devices. I wear a signal mirror and signal whistle on a paracord necklace around my neck. This way, if I'm ever in an emergency and can't move, I can still call for help effectively.
-shelter. For non-survival camping, I like a tent and sleeping bag with a foam sleeping mat. On day-hikes, I bring a tarp in my backpack, just in case I get caught in a downpour and need to quickly make a temporary shelter.
-food and water. If there is a water source, I'll take either a 2-micron absolute filter and use it for obtaining water or some Polar-Pur iodine drops. I've never had a problem when using either of those purification methods. For camping in North America, a 2-micron absolute filter is fine. For food, it depends on how close to the car I am. I like to take a cooler, filled with ice, when I'm near the vehicle and don't have to carry the cooler too far. Otherwise, I take non-perishable items, including lots of nuts for protein.
-tools. Ah, here's my favorite part

I never go into the wilderness without a strong fixed-blade knife. I know that, if need be, I can use it to build a shelter, shape traps and snares, prepare food/game, and make fire. Since it's impossible to predict a disaster, it's always wise to have some emergency gear with you. A knife is a fundamental tool, useful for all sorts of tasks. In addition to a knife, I like having a multi-tool; Leatherman is my favorite maker and Victorinox is also nice, from what I've heard. I don't care for Gerber's tools. I like a small, functional knife sharpener that I can wear on my belt, along with my multi-tool and camp knife. A small field saw can save a lot of time and wear-and-tear on your knife's edge. Lastly, take a good flashlight or headlamp with a set of extra batteries.
-fire. Fire is a basic and very important part of the wilderness experience. Fire can be used for so many different purposes, both emergency and non-emergency related. Physically, fire provides heat and light, both of which are very important to humans. In a traumatic circumstance, fire will also help a preson acquire a rational mindset. You can notice then when simply sitting by a fire - it is very calming and relaxing. Imagine how important that can be if you are having to deal with being lost, being snowed in, being stuck, having to care for a wounded person until help arrives, etc!
For basic fire tools, I take a metal match:
http://countycomm.com/FIRESTARTER.htm After you've practiced shooting a spark from the match, using your camp knife, you can start literally thousands of fires with this! It requires a knife/steel to strike the spark, and some tinder that can light with a spark. Examples of this would include: dry cotton balls, cotton balls smeared with vaseline/petroleum jelly (these will burn really well!), fatwood/pitchwood shavings (this is wood from any tree that contains resin, a turpentine-like substance. You locate said wood, shave it with your camp knife, strike a spark and you have fire!), magnesium shavings, and any number of pre-made fire starters available at outdoor stores.
Finally , on fires, I am not a fan or proponent of matches and lighters. Yes, they are better than nothing, but htey have some inherent limitations and weaknesses that make them undesireable for field use
One last bit of advice: make sure you know where you are going before you leave and that you leave detailed directions and an itinerary back at home with a close friend or relative, someone who will notice if you don't return on time.
I hope I haven't gone on too long; this is one of my very favorite subjects!

I think I've hit on what I consider to be most important. Tools and gear are great; the skill to use them properly is much more important!
Hope you have lots of fun, stay safe, and learn a great deal about life in the wilderness!
