When working within Camtasia, your files are saved as cmproj files. Cmproj* is a proprietary file that will not work outside of Camtasia.
When you have completed assembling your movie, you can use the Share button in the top right corner of the Camtasia interface as seen below. This opens up the various options available to you. For more information about each, see below the image.
Local File - This allows you to save an MP4 file directly to your desktop. This is recommended for a number of reasons, the biggest being that it keeps a stable file around for future use. It also allows you to upload the video to the site of your choosing even if it isn't depicted here (i.e. Dropbox).
Screencast.com - Screencast is the site maintained by Techsmith to support Camtasia. Since we do not have a Screencast account, this link has little use for us.
YouTube - YouTube is an easy place to host your content. You can log into your personal CSUF account to host videos specific to your needs. If you feel that there may be broader interest in the topic, you might want to consider hosting it on the Pollak Library eLearning channel. If that's the case don't hesitate to let me know and I'll be happy to upload it for you! As an added bonus, you can use YouTube's auto-captioning service when you upload.
Google Drive - Google Drive is another handy place to store videos, but it should be noted that Drive often has trouble embedding videos, and sometimes captioning can be lost. Drive is nice to use just as an archive, but isn't ideal as a viewing platform.
Share history - Allows you to see what you've previously created and where it might have been saved.
Hosting your video on a website where it can be easily accessed is a great idea. However, remember that if you upload it to a site like Google Drive, the video would likely have to be downloaded in order to be watched. Furthermore, although some sites might allow you to view the video directly, they often will have an effect on other features, such as captions.
I personally have used YouTube for the fact that it is easy, accessible, and has the added benefit of somewhat reliable auto-captioning.