The only way to move content via OOTB methods is by upgrading. There
are some 3rd party apps (like AvePoint) which will allow you to move
items in a more granular manner.

the OOTB methods are the way to go. I would probably
make a SQL backup of the content database (if I recall correctly, that's
the only one that really means anything, as the prof and config can be
re-created), then upgrade it in an dev environment just to make sure.

Just make sure that if you have customized webparts, deal with those
first.

As the others have said the upgrade process is the way to go but the
problem you may encounter is the look and feel of the sites may take
some additional effort to them to look like 2007 sites. Something to
consider is the CorasWorks design migrator tool. We have used it
quite a few times for both migration and mass rebranding of MOSS
sites.