Mattail has correctly pointed out many useful tools for helping accomplish this, but it’s that “databse driven stuff” that is likely to be the fly in the ointment.
Considerations such as the database engine in use (MySQL, Postgresql, DBM, etc., and their versions), the languages the database stuff is written in (version issues, available features, etc.), the actual code that drives the database stuff (standard and freely available web applications, and if so what versions, is it custom code, are paths “hardcoded”, are there easily modifiable configuration files to be tweaked, etc) and the OS/file structure in place at your previous host have everything to do with how big a chore this is likely to be.
Do you have shell access at the previous host? Can you post a url so that we can see what is involved?