Greg would be a good person to talk to about this... NS member Mr.Panda (PM him) he trains in martial arts, like daily, for a long time now. I know it isn't boxing but they're similar enough in how you get into it.
Anyway, I did muay thai kickboxing for awhile and enjoyed that a lot. Often times, boxing gyms or in my case muay thai gyms, would offer a free session so you could go and see what it's all about. I would recommend you find the closest boxing gym and see if they have a "teaser class" or something similar. If not, go visit the place and get prices and see what you think. If it looks pretty cool and you get all excited being there - give it a shot! People are super nice and encouraging at those places generally, so just walk right in and tell them you're new and interested in trying it and i'm sure they'll hook you up with info.
If you find that it's too expensive, buy a heavy bag as has been said, and some training gloves, and start learning on your own. The internet is a great place for learning about those types of things....watch youtube videos....google boxing workouts and routines....and get started that way.