Moving custom content types and content
Both Feeds and Migrate are modules purpose-built for what you'd like to do. Migrate requires a little more coding at this point, but I like its ease of use a little more than Feeds.

Simply upgrading the site, using CCK for Drupal 7 to migrate your fields and content types, might be a better solution, though. I would create a backup copy of your site and at least give it an hour's worth of trying before resorting to a full-on content migration.