If you're already pushing the limits of your share hosting account, I'd recommend starting with the 1GB Droplet and scaling as needed.
That being said, DigitalOcean is a self-managed platform, so unlike shared hosting, there's no control panel (i.e. cPanel) and setting up a Droplet (VPS) is either done from the CLI (Command Line Interface) or using a one-click installer (if available).
Going the CLI route, you'd be responsible for setting up your stack. Since you're using PHP, that stack would be either Apache, PHP, and MySQL/MariaDB or NGINX, PHP-FPM, and MariaDB. This could be done using the operating systems' package manager or by compiling each from source where using the package manager would be the faster option. Of course, this only handles installing the software needed to get those specific pieces working. You'd still need to do setup beyond the software installs, such as setting up directories configuring your web server to properly route your domains, etc.
With the one-click installer option you could setup a WordPress installation quickly and simply bring your data over and overwrite the existing installation, however, you'd need to migrate MyBB over and handle any setup required for it as the one-click images, while not limited specifically to what they are installing, are targeting specific users more or less (i.e. those who want to get up and running quick, but most likely don't have a need for multiple installations).
While I highly recommend DigitalOcean (I've used them for years, and continue to do so), if you're not too comfortable working from the command line, jumping from shared hosting where everything is more or less handled for you to and environment such as what is provided by DigitalOcean may be a little bit of a shock.