If you are going to Hostgator this is a snap. Read this article at Hostgator:
How to switch hosts without having any down time.
Note in particular these two points:
- Do not tell your old host! They could close down your site. Not ethical, but not unheard of.
- Look at point two in particular where it says that Hostgator will do it for you if you want. I would recommend taking them up on that.
In addition, I would personally recommend doing a simple file download of your complete site as a backup now. Also you need to download your database files for WordPress. This is a little bit slow because WordPress is a big program but I would recommend it.