It's all about having fun and coming back in one piece. Good for you for asking the question, that's always the first step to learning. "How much experience" is probably the wrong question, though - you need to match some skills to the situation and that's hard to calculate based upon X number of dives.
Wreck diving, even if you aren't going inside, can be far more dangerous than a simple dive on a reef or featureless bottom. Every fisherman knows that wrecks are great places to fish but that they are also easy places to snag a line and lose it. That lost line can mean serious entanglement hazards exist on lots of "easy" wreck dives that divers very often can't see until they're snagged. Add in the potential for deteriorating conditions that lead to unstable parts collapsing and a diver has at least a couple of ways to find himself in a tight spot without ever penetrating the wreck.
So long as you stay well off a wreck that is outside the zone where fishing line might snag you and nothing can fall over on top of you I don't know that any special skills are required. Once you start getting close, however, you need to have a presence of mind and a solid buddy, as well as the equipment and planning to deal with the possible emergencies. For instance, how do you untangle a buddy caught in monofilament (which can be nearly invisible) without getting caught up in it yourself?
Actual penetration of a wreck is probably the most dangerous diving that the average person will ever do. Foot for foot, penetrating a wreck is more dangerous than cave diving due to the potential for disorientation, silt-outs, snags, collapses, cutting and poking edges, swinging and toppling everything, etc. Except for currents inside the wreck, you've got all the hazards of cave diving plus a bunch of new ones, so getting prepared is critical.
I'm old-school enough to believe that every diver should have the goal of getting Rescue certified. It's been my experience that Rescue is the class where divers are faced with situations and ideas that will push them to learn to be self-reliant and independent. Even if you aren't going to be wreck diving, Rescue should be on your list of classes to take - but if you're planning to get close to any kind of wreck, Rescue should be mandatory. Beyond that, skills and gear need to be matched to the wreck and to your plan. You need to be comfortable in the water and have mastered (and routinely practice) all of the basic skills: multiple kinds of finning techniques, mask removal and replacement, alternate air sharing, and more, depending upon the dive. You need to be able to control your buoyancy well enough to hold a position without moving your arms or feet so that you won't be stirring up the silt. You need to have good signaling skills so that you and your buddy can communicate effectively. I do a lot of wreck diving and never - NEVER - go inside one without a completely redundant gas supply and multiple spools, lights and cutting devices. PADI has a lightweight Wreck class that is good for those who are just looking to swim around the wreck and don't intend to go inside, full penetration requires far more serious training. Cave diving skills translate pretty well to wreck diving, so most of the Cavern and Cave classes taught by several agencies are going to teach you things you need to know.
Have a plan and take it one step at a time and you'll find yourself deep inside a wreck in good enough time. Enjoy the adventure and remember that the journey to your destination should be half the fun.
