It would be best to grab a copy of the website files/code base as well in almost every case.
For example the code that manipulates the look of the site is contained in the file structure.
Most of the text that is found on pages are in the sql DB.
The way content is displayed on some pages and not others could be in DB or the way the files manipulate the DB.
Therefor the standard practice is to take a snapshot of the code base as well as the database, as a pair, to the new hosting environment.
I would strongly suggest that you have a really good look at the D7 migration manual. It may look lengthy, but there are so many things that could derail your move, it's definitely worth it to know what's going on if things go funny.