In theory, it goes like this.

I take a backup of the 'site' (all the files, images and structure of the current site)

The backup gets transferred to the new server.

Some files need to be modified to reflect the new server settings.

I then take a backup copy of the database and transfer it to the new server.

I create a new database on the new servers and restore the backup into it.

At that point the new server can be tested for a few days.

Once everything is verified as working, and depending on how long it takes, I will either close the old forum (or post a message saying that anything new will not be tranferred)

backup the database again, tansfer it, restore it and open the site on the new server.

I'll then get the DNS records changed and put a redirect on the old site, pointing at the new one.

Job jobbed.