Hi,
It all depends on what your personal requrements are, like, do you want a powerful/fast server or just something to start off?. I think another thing to take into consideration is "how much do you know" i say that because you can get Dedicated servers that YOU run or you can get Managed servers, this is where the datacenter install software for you etc etc. I would recommend starting small, go for a managed server, that way if anything goes wrong it can be soerted out pretty much right away, then once you gain more and more experience, move up a level to a bigger server. Once you have enough confidence then you can get a self-managed server. As for hosting from your own computer, that is a no no in my book, because your alot more prone to attacks also, what if you have a power outage? and you need a very fast connection, i think the lowest you could go is T1 connection.
I hope this helps you.
Thank you
Sin