Change hosting server for website

I would like to move a currently operating website from one server to a new one with a different company. Currently the site, which is small and has little traffic, is hosted by a small specialised company (Krome) and I would like to move it to Dreamhost.

I have been waisting hours trying to move it over but as I am not an IT expert I am struggling as to date all I have managed to do is point the domain at the new server, but I've been unable to upload the SQL file. As a result the site is currently down, in internet limbo, and we need it up and running ASAP.

We are the new company have been very helpful so far but it's still waisting too much of my time and I don't want to make a mess of it all. Please help!