Firstly, running your board successfully, you will need to consider the server configuration as phpBB forums takes up resources, RAM & CPU. For this purpose we use high end configurations shared servers with properly tuned.
For your bulletin board you will require MySQL database with PHP and considering your requirement any of our superior Linux shared hosting plan will be fine. You can install it yourself through the control panel or can ask our technical member to get it done for you. phpBB board is easy to set up, administer and absolutely free.
Growth of the forum depends on the theme and reputation of the forum which will ultimately decide the exact plan to upgrade (VPS / Dedicated Hosting) from shared hosting. As the visitors to your forum increases, the load may go over the limit of the shared servers.