Dave,
One of the primary benefits of being on a dedicated server is running scripts and other services that may be processor intensive. But remember, unless you are going to pay for managed support, you are the root admin when you have your own server. You are in control of the amount of disk space you use, the email server, dns, etc...
The other benefit to having your own dedicated server is you are not at the mercy of the hosting provider where most have upwards of 150-200+ sites on a single server, you will be the only one on your box. It will also let you run as many websites as you want without incurring additional costs except for bandwidth.
As far as how and who, your best bet is to talk with your provider and let them know you are looking to get a dedicated server or check out sites like tophosts.com and click on dedicated.
As for cost, dedicated servers will start around $99.00/month and can go into the thousands depending on the amount of hardware and bandwidth you require.
Hope this helps.
Mike Snetzko
MJSMedia/worldbizhosting.com