Proper method for transfering Domain/Hosting to DreamHost (WordPress)

I have a company site hosted on I need to transfer both the hosting and email over to DreamHost. I already have a VPS on DreamHost set up. I am looking for the proper path to take to minimize downtime of the website and email services since our company relies on its web presence email service.

Can someone please tell me the proper steps to take so that there minimal to no downtime? I have seen the wiki about transferring sites but when I tried that (maybe I did something wrong) with my personal site, there was 3-4 days when it was down. Not a huge thing for me, but for my company that is big.

Would it be best to buy a new domain name on DH, then change the nameservers on Startlogic to point to the new domain on DH, Then move all the emails over? I am lost as to the best method of doing this. If I have to pay extra, that is fine.