Theres really nothing to getting started. The trick is and important thing is to learn to read the weather data, the storms themselves, the clouds, stuff like that. You dont necessarily have to be a meteorologist but it does help to have a good understanding of weather, its essential to your safety. I pay a fee each month to be tied into the National Weather Service's data so I can have the radar information, warning information, lighting data and so forth with minimal time delay. Ive chased storms very early on without it, most notably the May 3, 1999 outbreak here in OK and nearly had an F3 tornado land right on top of me because I was blind to what was going on.
If they offer any type of storm spotter classes in your area (they are usually free) thats a very good place to start. Then its a matter of what risk you are willing to take, getting some good camera gear and willingness to go without sleep and drive forever and be okay when you come away with nothing.
I chase storms mainly for the lightning. And I get really, REALLY close to them but thats a risk Im comfortable with. Ive also left work with my camera gear packed, laptop and all the other good stuff (back when I shot film I usually have 20 or so rolls of high end film with me too) went out chased storms from the time I left work until the wee hours of the morning, pulled back into the parking lot of work, got maybe an hour or two of sleep and went back to work again.
But when you come away with that one picture, it makes it worth while to me.