Does hosting serverses offer your own .com or do you have to get it yourself?

Alot of hosting companies do sell domain names in addition to just hosting. Depending on the Host and the package's they offer, you may find some hosts offering special deals with the purchase of a plan with them (i.e 1 years free domain registering, 6 months, etc...).

I've found that it's best to buy your domain apart from your hosting though. That way, if you need to switch hosting, it's easier to continue to manage your domain, and your host can't pull the plug.

Wahoo is totally right. Even if your host offers to buy your domain, say no unless they are willing to give you full control. Many hosts will hold on to your domain to prevent you from leaving, forcing you to deal with exorbitant prices and/or shoddy service.

Conker

Even if your host offers to buy your domain, say no unless they are willing to give you full control.

I think that you shouldn't worry about that - as almost all web hosts provide only interface between them and domain name registrar. All data you provide by yourself only

Any web sites where i can register my domain?

GoDaddy.Com is highly recommended. I also recommend RegisterSite.Com. Most people don't know about RS, but I have had great service with them.