Well said on the title. I don't know where should I ask this question tho. Please tell me the right place if I'm wrong.
If your forum is new, it will not get that much traffic. You can go for free hosting for now. Try freehostia.com or any economy plans of the hosting companies such as godaddy.com, rackspider.com, etc.
You need to have good shared web hosting which supports MySQL and PHP you need. As rule forum developers have a list of the web hosting recommendations