I would go with a dedicated server that is located in an established datacenter.
Setting this up in your home, making it efficient, and providing quality service is going to cost a pretty penny.
At the least, you'll end up having to purchase a T1 line from your phone company, in order to actually run the service.
For example, Verizon DSL (which I use at home, and we use Verizon Business DSL in our office) will NOT allow you to run a web server from the DSL line, but will allow you to run a server from their T1 lines. The problem here is, it's going to cost quite a bit to order the T1 line, and then in most areas where a line is not readily available to tap into to fork off to your location, they're going to need to install it (dig, lay line, etc)....guess who gets the bill for that.
Now take into account the cost of buying the server rather than renting it. You're looking at anywhere from a few hundred dollars to $2k to buy a quality server straight out, depending on specs needed and brand.
At this stage in the game, I'm assuming you're still somewhat of a novice, which is cool, but I wouldn't drop a bucket of cash like that, until you learn a bit more about it, etc.
I'd definately go with renting a server from a reputable datacenter or server provider.