After you saved your site as a template, go to the Solution Gallery (Go to top parent site, then Site Actions > Site Settings > Solutions (under Galleries))
You'll see your site template here. Click the name of the site template. This will let you download the file.
Now go to your other farm, and go to Solutions (same path as above) and upload the file. To upload, you'll have to click Solutions tab again in the ribbon. For some reason it shows the Browse tab in the Solutions tab on initial page load.
Upload the solution, then activate it. Then it should be available when creating new sites.