There are a lot of ways you can take this, and a lot depends on the kind of traffic that you expect to receive. What are the specs on the VPS you'll be migrating from?
I'm partial to Nginx, but there's no reason not to use Apache if you are more comfortable with it. While Nginx does use slightly less resources, Apache is probably not your real problem. It's much more likely to be MySql.
The first thing is to make sure you are on a server with enough RAM. For example, running an entire website on a 1GB droplet will very likely be more performant than running the front end on a 512MB droplet and the database on a separate 512MB droplet, especially if the problem is MySql's memory usage.
Once you get to a larger scale, running the database on a separate server may begin to make sense. Check out this article:
It uses Wordpress as an example, but will mostly apply to phpBB as well.
Let us know how it goes, and feel free to ask questions if you hit on a specific problem.