OK, so maybe you do need a server.

I've never had a server of my own, just played around a little on a couple. If it were me, I would absolutely go with a dedicated server. I don't see much advantage in actually owning the hardware unless you build your custom boxes yourself or something. Of course, if you can hook your machine(s) up at a data center near you and are allowed to go there in person if need it may be different. But it's also one more thing that can go wrong.
I haven't visited webhostingtalk in years, but it used to be a place where you could get advice about these kind of things (and offers too). Not entirely sure that is true anymore. Last I was there it seemed to be a little too much "promotion" for my taste going on. But give it a go.
http://webhostingtalk.comIf you find a host you like that offers both shared hosting and servers, I think you should discuss your plans with them. They can probably tell you if you will be OK on a shared hosting account or a VDC or if you need a whole server.