As for most outdoor activities, you can certainly get by with non-specialized clothing for skiing and snowboarding. As @imsodin points out in the comments, a ski resort is (compared to a lot of outdoor settings) a pretty safe place to make a mistake about clothing: if you find yourself too cold or wet, you can get indoors quickly; if you stay on-piste, your chances of death or serious injury from exposure are practically nil. Another advantage is that you're only skiing for, usually, a few hours per day; if your clothes get soaked, they have >12 hours to dry out overnight before you go back out.
Ski-field conditions can vary hugely in terms of sun, wind, air temperature, and precipitation -- even during the course of a day. Whatever you wear needs to be able to deal, to some extent, with any conditions you're likely to encounter. If there's a chance you'll hit freeze-thaw conditions, you'll want a jacket, and preferably trousers, that's reasonably waterproof. You're unlikely to find yourself skiing in pouring rain, so serious full-on waterproofing isn't essential, but your jacket should at least be able to handle a bit of sleet and wet snow without soaking through.
I skied for years without any specialist clothing, though admittedly only for a week or so per season. Initially I was using a cheap, sort-of-slightly-water-resistant jacket and overtrousers, sized fairly large, with as many layers of synthetic thermals and fleeces as required to keep me warm on the day. For my first time skiing or snowboarding, I definitely wouldn't invest in a specialized jacket. Even if you end up buying one later, you'll be able to make a better selection once you have some experience on the slopes. There's no question that a specialized skiing/snowboarding jacket makes life more pleasant, but it's by no means necessary.
As a slight aside, there are three things you will want to think about: gloves, goggles, and a helmet. Ski gloves are thicker than most everyday gloves, and layering is seldom effective: you can get away with a light glove liner and generously sized outer glove, but in general you risk reducing your circulation and making your hands colder if you try to stack up too many gloves (source: bitter experience). Goggles are also hard to substitute with everyday equivalents: in good conditions you can get by with a decent pair of wrap-around sunglasses, but even cheap goggles do a lot better as soon as you have wind, snow, or sweat to contend with. As for the helmet, you can probably hire one, but check the prices: often it doesn't take many days of rental charges to equal the purchase price.
