I thinking setup a Web Site and I was thinking is: Would I need a domain name or hosting or both?

[AnnOzerguy]

schweppesy has got a very good point; don't put the cart in front of the horses. Managing your own server is no small matter. You need the server, the storage, the backup storage, the permanent access to the internet, backup access to the internet, backup power, the softwares (web server, email, firewall, php, mysql, etc.), advanced knowledge of everything in order to keep your system up to date, as safe as possible, etc.

Even after being in this business for 7+ years (with 50+ websites), I don't run my own server but have a fully managed VPS (fully managed meaning that the host actually manages the server on my behalf). I started off with a shared hosting account at hostgator, later upgraded to a reseller account (so I could have 1 cpanel per site, and give full cpanel access to my outsourcers for the 1 site I wanted them to work on), and finally got a fully managed VPS. I would not run my own server from my basement, for example. My next upgrade shall be a fully managed dedicated server.

My recommendation:
- get a domain name from namecheap
- get a baby shared hosting account from hostgator
and you should be fine for a little while...