Stick welding is about the cheapest to get started in if you really want to have any strength to what your doing. A little training and a lot of practice are required. Your also working with a 3000 degree pool of molten steel, burning flux, and strong ultra violet light. Not something to do inside your house
I'd suggest you head over to the shoptalk section
http://www.bladeforums.com/forums/forumdisplay.php?f=741
And read through all the recent welding threads, there 3 or 4 that have come up recently and include all kinds of general info as well as lots of advice on what kind of welder to purchase.
A good oxy acteylene rig will do a heck of a lot(cutting, welding, brazing and soldering) but is going to set you back as much as a good stick welder would, and would not be safe to store or use in your house. It also requires some very specific training, because its a lot more involved than turning a few knobs and lighting it up. You could very easily kill yourself and anyone nearby if you do something wrong.
In fact, I can't think of any welding operation that would really be safe to do in your house or on a balcony. There's just too much heat coming off, too many fumes coming off, and too much molten metal and slag flying around. It really has to be done in some sort of shop or garage where you can contain everything, and definitely not overhead of other people!

If you can even get out in the open it would probably be ok, but you have to be careful so that your not starting fires or anything like that.
I'd say your best bet would be to take a class of some sort, there are even some offered that cover artisitic metal work. That way you can get trained in safe operation, and get an idea of what kind of facilties are necessary. You also would get a chance to try a few different welding processes out and see which one you might be interested in investing in.