Migrating from MyBB or another piggy-back host
You will need a copy of your database with CREATE TABLE statements and a copy of your FTP files. This is the most seamless of moves.
Migrating from IcyBoards
Arrange with Spork (Icyboards owner) to export your database with CREATE TABLE statements. You should arrange this so that when it is exported, all of your plugins are turned off.
Check with me for specific details. Please note that you do not have to use a MyBB forum, it just happens to be what most of my hostees do.
My policy: you own it, it's yours
If I purchased a domain name for you or transfered yours to my account, billing would be a nightmare. Instead, you own and manage your own domain name. Instead, I help you point your domain to your new home. Typically domain names run $10 USD/year. I highly recommend purchasing Privacy Protection (about $10USD/year), as it hides your real, personal information from malevolent people as well as web-robots looking to harvest it and sell it to the highest bidder.
To point the domain that you own and manage yourself to your new website, follow these instructions (here), or you can use this article (here) as it goes over the same thing. You will want to use these two nameservers, same as in the first article:
This change usually takes a couple of days as the internet learns your new address (so you will want to change your nameservers as soon as possible). Once the change has taken effect, let me know, and I will go into my account to complete the link, and also to change your cookie settings so that everyone will be able to log in/out with your new URL.