What blog software are you using? Most blog software has some sort of backup function that you can use.
Also, if you are switching to similar hosting, there are ways to copy an entire site. For instance, if you are switching from a cpanel hosting account to another cpanel hosting account, it can be fairly straightforward to copy any site.
In any event, if none of the above applies to you, you just need to copy all the files from your old host to your hard drive, then from your hard drive to your new host. You will also need to copy the database onto your hard drive then upload that to your new host. You may need to change some settings at your new host if you have changed the database name.