It depends on your definition of hosting yourself. Most people start at renting server space from a hosting company like hostgator. This gives you access to your "server" which is really shared space.
With that said, you are the administrator of your "space". You control your wordpress and have access to some functions on the server using a control panel, you can get away with not having to mess with the server end too much until you learn to use it or you can give access to someone you've hired to do the work for you if you so choose (your decision). You can also set up a user for the server for web work you hire out, then turn off when job is complete. Host gator will run you about 5-15 dollars a month.
The next step would be called a VPS (this is what I run) It stands for virtual private server, which segments off space for you so there are less people on your "box". I also sell space to clients who need web design. I'm thinking this is not what you are referring to when you say server. This will run you close to 100.00 month +. If you just run a website, you want the above option.
The step after that is called a dedicated box or this is in the truest sense your own server. This will run you more, but you are the only one on that server (unless you give access to that machine. This is beyond your scope, i believe.
What is your current url? to get a "sence of where you're at"...
Last edited by Lil Chris; 11-22-2011 at 02:14 AM.