Hosting a business on which one relies for income on a shared server would be considered an initial point of failure. Query your IT department about who made the decision to use a shared resource and instruct them to research a more suitable solution. And then fire them.
The Wiki has lots of good info, and there is this forum.
You'd first attain database dumps (backups) of your current databases and then import them here.
There is a brief description of the transfer process outlined in the Wiki: http://wiki.dreamhost.com/Transfer_your_hosting_to_us
Pick the situation that applies to you and it's reasonably straight forward
If you're moving everything over, tell Support (they might even offer to do it for you considering you're changing over to DH). Ask them if they will - it's worth a shot!