There are many ways of exporting and importing taxonomies and terms in Drupal, but one of the more simpler ways is to use the Taxonomy CSV import/export module. It lets you choose one of many taxonomies, like the tag taxonomy for example, export it to a comma-separated list which you then can import in a similar way.
In this video I'll show you how to do an export from one Drupal installation, in this case a Drupal 6 version (just to show that it's possible to move tags between different Drupal versions as well) and then import it into another Drupal installation (Drupal 7 in this case).
It all starts with the export where you choose which vocabulary/ies you want to export and how you want to format it.
The second stage is to import the CSV-file into your target installation, and I'll show you this as well.
The module has Drush support as well, so if you use Drush much you might want to use the terminal for your export and import.