Drupal site needs maintenence

Site is still operating, but doesn't update satisfactorily.
Was working until recent host migration, now seems somewhat broken, each page when logged in as admin shows: "warning: date() [function.date]: It is not safe to rely on the system's timezone settings...." about 20 times.
Can't update pages because they lose images when edited, then require the images.

Would like site to be fully working.
Would like site updating to newest Drupal (on 6.15) for security.
Would also be great to have a sort out. Currently at least two copies of necessary files uploaded and lots of other legacy junk, would like to recover space.

Don't see this as a big job, but would be keen to have attention to detail, full backups, etc.

Open to ideas following this work for improvements to the site, ways to allow more content up there etc.